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# 4. Синус, косинус, тангенс, котангенс — теория по математике

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```1.4. Синус, косинус, тангенс, котангенс
1.4.1. Понятие синуса, косинуса, тангенса и котангенса
числового аргумента
&ETH;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &aring;&auml;&egrave;&iacute;&egrave;&divide;&iacute;&oacute;&thorn; &icirc;&ecirc;&eth;&oacute;&aelig;&iacute;&icirc;&ntilde;&ograve;&uuml;, &ograve;. &aring;. &icirc;&ecirc;&eth;&oacute;&aelig;&iacute;&icirc;&ntilde;&ograve;&uuml; &ntilde; &ouml;&aring;&iacute;&ograve;&eth;&icirc;&igrave; &acirc; &iacute;&agrave;&divide;&agrave;&euml;&aring; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve; &egrave; &eth;&agrave;&auml;&egrave;&oacute;&ntilde;&icirc;&igrave; 1.
&Ntilde;&egrave;&iacute;&oacute;&ntilde;&icirc;&igrave; &divide;&egrave;&ntilde;&euml;&agrave; α &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&agrave; &ograve;&icirc;&divide;&ecirc;&egrave; &ETH;α, &icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&acirc;&agrave;&iacute;&iacute;&icirc;&eacute; &iuml;&icirc;&acirc;&icirc;&eth;&icirc;(x;
y)
P
&ograve;&icirc;&igrave;
&ograve;&icirc;&divide;&ecirc;&egrave; P0(1; 0) &acirc;&icirc;&ecirc;&eth;&oacute;&atilde; &iacute;&agrave;&divide;&agrave;&euml;&agrave; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve; &iacute;&agrave; &oacute;&atilde;&icirc;&euml; α &eth;&agrave;&auml;&egrave;&agrave;&iacute;.
α
y
&Icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&agrave;&aring;&ograve;&ntilde;&yuml;: sin α, &ograve;. &aring;.
α
0
P0
x 1
sin α = y — &icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&agrave; &ograve;&icirc;&divide;&ecirc;&egrave; Pα.
&Ecirc;&icirc;&ntilde;&egrave;&iacute;&oacute;&ntilde;&icirc;&igrave; &divide;&egrave;&ntilde;&euml;&agrave; α &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &agrave;&aacute;&ntilde;&ouml;&egrave;&ntilde;&ntilde;&agrave; &ograve;&icirc;&divide;&ecirc;&egrave; Pα, &iuml;&icirc;&euml;&oacute;&divide;&aring;&iacute;&iacute;&icirc;&eacute; &iuml;&icirc;&acirc;&icirc;&eth;&icirc;&ograve;&icirc;&igrave; &ograve;&icirc;&divide;&ecirc;&egrave; P0(1; 0) &acirc;&icirc;&ecirc;&eth;&oacute;&atilde; &iacute;&agrave;&divide;&agrave;&euml;&agrave; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve; &iacute;&agrave; &oacute;&atilde;&icirc;&euml; α &eth;&agrave;&auml;&egrave;&agrave;&iacute;.
&Icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&agrave;&aring;&ograve;&ntilde;&yuml;: cos α, &ograve;. &aring;.
cos α = x — &agrave;&aacute;&ntilde;&ouml;&egrave;&ntilde;&ntilde;&agrave; &ograve;&icirc;&divide;&ecirc;&egrave; Pα.
&Ntilde;&egrave;&iacute;&oacute;&ntilde; &egrave; &ecirc;&icirc;&ntilde;&egrave;&iacute;&oacute;&ntilde; &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&ucirc; &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&atilde;&icirc; &divide;&egrave;&ntilde;&euml;&agrave; α.
|sin α| ≤ 1, |cos α| ≤ 1.
&Ograve;&agrave;&iacute;&atilde;&aring;&iacute;&ntilde;&icirc;&igrave; &divide;&egrave;&ntilde;&euml;&agrave; α &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&aring; &ntilde;&egrave;&iacute;&oacute;&ntilde;&agrave; &divide;&egrave;&ntilde;&euml;&agrave; α &ecirc; &aring;&atilde;&icirc; &ecirc;&icirc;&ntilde;&egrave;&iacute;&oacute;&ntilde;&oacute;:
tg α =
sin α
.
cos α
π
+ πn, n ∈ Z.
2
&Ecirc;&icirc;&ograve;&agrave;&iacute;&atilde;&aring;&iacute;&ntilde;&icirc;&igrave; &divide;&egrave;&ntilde;&euml;&agrave; α &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&aring; &ecirc;&icirc;&ntilde;&egrave;&iacute;&oacute;&ntilde;&agrave; &divide;&egrave;&ntilde;&euml;&agrave; α &ecirc; &aring;&atilde;&icirc;
&ntilde;&egrave;&iacute;&oacute;&ntilde;&oacute;:
&Ograve;&agrave;&iacute;&atilde;&aring;&iacute;&ntilde; &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute; &auml;&euml;&yuml; &acirc;&ntilde;&aring;&otilde; α, &ecirc;&eth;&icirc;&igrave;&aring; &ograve;&aring;&otilde; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&egrave;&eacute;, &iuml;&eth;&egrave; &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&otilde; cos α = 0, &ograve;. &aring;. α =
y
O
Pα (x; y)
R
α
P0
ctg α =
x
cos α
.
sin α
&Ecirc;&icirc;&ograve;&agrave;&iacute;&atilde;&aring;&iacute;&ntilde; &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute; &auml;&euml;&yuml; &acirc;&ntilde;&aring;&otilde; α, &ecirc;&eth;&icirc;&igrave;&aring; &ograve;&aring;&otilde;, &iuml;&eth;&egrave; &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&otilde; sin α = 0,
&ograve;. &aring;. α = πn, n ∈ Z.
&Auml;&euml;&yuml; &icirc;&ecirc;&eth;&oacute;&aelig;&iacute;&icirc;&ntilde;&ograve;&egrave; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&icirc;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &eth;&agrave;&auml;&egrave;&oacute;&ntilde;&agrave; R &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&yuml; &ograve;&eth;&egrave;&atilde;&icirc;&iacute;&icirc;&igrave;&aring;&ograve;&eth;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&otilde; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&eacute; &acirc;&ucirc;&atilde;&euml;&yuml;&auml;&yuml;&ograve; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&egrave;&igrave; &icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&igrave;:
sin α =
y
;
R
cos α =
x
;
R
tg α =
y
;
x
ctg α =
x
.
y
&Aring;&ntilde;&euml;&egrave; n ∈ Z, &ograve;&icirc; &ntilde;&iuml;&eth;&agrave;&acirc;&aring;&auml;&euml;&egrave;&acirc;&ucirc; &eth;&agrave;&acirc;&aring;&iacute;&ntilde;&ograve;&acirc;&agrave;:
sin (α + 2πn) = sin α; cos (α + 2πn) = cos α.
&Aring;&ntilde;&euml;&egrave; n ∈ Z, &ograve;&icirc; &ntilde;&iuml;&eth;&agrave;&acirc;&aring;&auml;&euml;&egrave;&acirc;&ucirc; &eth;&agrave;&acirc;&aring;&iacute;&ntilde;&ograve;&acirc;&agrave;:
tg (α + πn) = tg α; ctg (α + πn) = ctg α.
32
Раздел 1. Выражения и преобразования
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 1. &Iacute;&agrave;&eacute;&auml;&egrave;&ograve;&aring; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&egrave;&aring; &acirc;&ucirc;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml;:
π
π
π
π
⎛ π⎞
⎛ π⎞
&agrave;) tg ⎜ − ⎟ ⋅ ctg + sin + 2 cos ⎜ − ⎟ = −tg ⋅ 1 + 1 + 2 cos =
4
2
4
6
⎝ 6⎠
⎝ 4⎠
= −1 ⋅ 1 + 1 + 2 ⋅
3
= 3;
2
&aacute;) sin 405&deg; + ctg 570&deg; = sin (360&deg; + 45&deg;) + ctg (540&deg; + 30&deg;) = sin 45&deg; + ctg 30&deg; =
2
+ 3.
2
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 2. &Icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&egrave;&ograve;&aring; &ccedil;&iacute;&agrave;&ecirc; &acirc;&ucirc;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml;:
&agrave;) cos 155&deg; ⋅ sin 255&deg; &gt; 0, &ograve;. &ecirc;. 155&deg; — &oacute;&atilde;&icirc;&euml; II &divide;&aring;&ograve;&acirc;&aring;&eth;&ograve;&egrave;, &ograve;&icirc;
cos 155&deg; &lt; 0; 255&deg; — &oacute;&atilde;&icirc;&euml; III &divide;&aring;&ograve;&acirc;&aring;&eth;&ograve;&egrave;, &ograve;&icirc; sin 255&deg; &lt; 0;
&aacute;) tg 127&deg; ⋅ ctg 201&deg; &lt; 0, &ograve;. &ecirc;. 127&deg; — &oacute;&atilde;&icirc;&euml; II &divide;&aring;&ograve;&acirc;&aring;&eth;&ograve;&egrave;, &ograve;&icirc;
tg 127&deg; &lt; 0; 201&deg; — &oacute;&atilde;&icirc;&euml; III &divide;&aring;&ograve;&acirc;&aring;&eth;&ograve;&egrave;, &ograve;&icirc; ctg 201&deg; &gt; 0.
1.4.2. Соотношения между тригонометрическими функциями одного аргумента
Основное тригонометрическое тождество
&Icirc;&ntilde;&iacute;&icirc;&acirc;&iacute;&icirc;&aring; &ograve;&eth;&egrave;&atilde;&icirc;&iacute;&icirc;&igrave;&aring;&ograve;&eth;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&aring; &ograve;&icirc;&aelig;&auml;&aring;&ntilde;&ograve;&acirc;&icirc;:
sin2 α + cos2 α = 1.
&Ntilde;&euml;&aring;&auml;&ntilde;&ograve;&acirc;&egrave;&yuml;:
sin2 α = 1 − cos2 α ⇒ sin α = &plusmn; 1 − cos2 α ;
cos2 α = 1 − sin2 α ⇒ cos α = &plusmn; 1 − sin2 α ;
sin α
π
&iuml;&eth;&egrave; α ≠ + πn, n ∈ Z;
tg α =
cos α
2
cos α
&iuml;&eth;&egrave; α ≠ πn, n ∈ Z;
ctg α =
sin α
π
1
1
tg α ⋅ ctg α = 1 &iuml;&eth;&egrave; α ≠ n, n ∈ Z; tg α =
, ctg α =
.
2
ctg α
tg α
1
π
&iuml;&eth;&egrave; α ≠ + πn, n ∈ Z.
1 + tg2 α =
2
2
cos α
1
2
&iuml;&eth;&egrave; α ≠ πn, n ∈ Z.
1 + ctg α =
sin2 α
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 1. &Igrave;&icirc;&atilde;&oacute;&ograve; &euml;&egrave; &icirc;&auml;&iacute;&icirc;&acirc;&eth;&aring;&igrave;&aring;&iacute;&iacute;&icirc; &aacute;&ucirc;&ograve;&uuml; &ntilde;&iuml;&eth;&agrave;&acirc;&aring;&auml;&euml;&egrave;&acirc;&ucirc; &eth;&agrave;&acirc;&aring;&iacute;&ntilde;&ograve;&acirc;&agrave;:
1
1
&agrave;) cos α =
&egrave; sin α = ?
2
2
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;. &Ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; &eth;&agrave;&ntilde;&ntilde;&igrave;&agrave;&ograve;&eth;&egrave;&acirc;&agrave;&thorn;&ograve;&ntilde;&yuml; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave; &ntilde;&egrave;&iacute;&oacute;&ntilde; &egrave; &ecirc;&icirc;&ntilde;&egrave;&iacute;&oacute;&ntilde; &icirc;&auml;&iacute;&icirc;&atilde;&icirc; &egrave; &ograve;&icirc;&atilde;&icirc; &aelig;&aring; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&agrave;, &ograve;&icirc;
&auml;&icirc;&euml;&aelig;&iacute;&icirc; &acirc;&ucirc;&iuml;&icirc;&euml;&iacute;&yuml;&aring;&ograve;&ntilde;&yuml; &icirc;&ntilde;&iacute;&icirc;&acirc;&iacute;&icirc;&aring; &ograve;&eth;&egrave;&atilde;&icirc;&iacute;&icirc;&igrave;&aring;&ograve;&eth;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&aring; &ograve;&icirc;&aelig;&auml;&aring;&ntilde;&ograve;&acirc;&icirc;:
2
2
1 1 1
⎛1⎞
⎛1⎞
cos2 α + sin2 α = 1, &iacute;&icirc; ⎜ ⎟ + ⎜ ⎟ = + = ≠ 1.
4 4 2
⎝2⎠
⎝2⎠
1
1
&egrave; sin α =
&icirc;&auml;&iacute;&icirc;&acirc;&eth;&aring;&igrave;&aring;&iacute;&iacute;&icirc; &ntilde;&iuml;&eth;&agrave;&acirc;&aring;&auml;&euml;&egrave;&acirc;&ucirc; &aacute;&ucirc;&ograve;&uuml; &iacute;&aring; &igrave;&icirc;&atilde;&oacute;&ograve; (&ograve;. &ecirc;. &iacute;&aring; &acirc;&ucirc;2
2
&iuml;&icirc;&euml;&iacute;&yuml;&aring;&ograve;&ntilde;&yuml; &icirc;&ntilde;&iacute;&icirc;&acirc;&iacute;&icirc;&aring; &ograve;&eth;&egrave;&atilde;&icirc;&iacute;&icirc;&igrave;&aring;&ograve;&eth;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&aring; &ograve;&icirc;&aelig;&auml;&aring;&ntilde;&ograve;&acirc;&icirc;).
&Iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute; &eth;&agrave;&acirc;&aring;&iacute;&ntilde;&ograve;&acirc;&agrave; cos α =
3
1
&egrave; cos α = −
?
2
2
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;. &Iuml;&eth;&icirc;&acirc;&aring;&eth;&egrave;&igrave; &acirc;&ucirc;&iuml;&icirc;&euml;&iacute;&aring;&iacute;&egrave;&aring; &icirc;&ntilde;&iacute;&icirc;&acirc;&iacute;&icirc;&atilde;&icirc; &ograve;&eth;&egrave;&atilde;&icirc;&iacute;&icirc;&igrave;&aring;&ograve;&eth;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&atilde;&icirc; &ograve;&icirc;&aelig;&auml;&aring;&ntilde;&ograve;&acirc;&agrave;:
2
2
⎛
3⎞
1 3 4
⎛1⎞
sin2 α + cos2 α = 1; ⎜ ⎟ + ⎜ −
⎟⎟ = + = = 1.
⎜
2
2
4
4 4
⎝ ⎠
⎝
⎠
&aacute;) sin α =
1.4. Синус, косинус, тангенс, котангенс
33
&Icirc;&ntilde;&iacute;&icirc;&acirc;&iacute;&icirc;&aring; &ograve;&eth;&egrave;&atilde;&icirc;&iacute;&icirc;&igrave;&aring;&ograve;&eth;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&aring; &ograve;&icirc;&aelig;&auml;&aring;&ntilde;&ograve;&acirc;&icirc; &acirc;&ucirc;&iuml;&icirc;&euml;&iacute;&yuml;&aring;&ograve;&ntilde;&yuml;. &Ccedil;&iacute;&agrave;&divide;&egrave;&ograve;, &eth;&agrave;&acirc;&aring;&iacute;&ntilde;&ograve;&acirc;&agrave;, &auml;&agrave;&iacute;&iacute;&ucirc;&aring; &acirc; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&egrave;,
&icirc;&auml;&iacute;&icirc;&acirc;&eth;&aring;&igrave;&aring;&iacute;&iacute;&icirc; &ntilde;&iuml;&eth;&agrave;&acirc;&aring;&auml;&euml;&egrave;&acirc;&ucirc;.
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 2. &Iacute;&agrave;&eacute;&auml;&egrave;&ograve;&aring; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&egrave;&yuml; &ograve;&eth;&egrave;&atilde;&icirc;&iacute;&icirc;&igrave;&aring;&ograve;&eth;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&otilde; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&eacute; &divide;&egrave;&ntilde;&euml;&agrave; α, &ccedil;&iacute;&agrave;&yuml;, &divide;&ograve;&icirc; sin α = 0,6
π
&egrave;
&lt; α &lt; π.
2
π
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;. &Ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; &iuml;&icirc; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&thorn;
&lt; α &lt; π, &ograve;&icirc; α — ∈ II &divide;&aring;&ograve;&acirc;&aring;&eth;&ograve;&egrave;. &Iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute;
2
cos α = − 1 − sin2 α = − 1 − 0, 62 = − 1 − 0, 36 = − 0, 64 = −0, 8;
sin α
0, 6
6
3
tg α =
=
=− =− ;
cos α −0, 8
8
4
1
1
4
1
ctg α =
=
= − = −1 .
3
tg α
3
3
−
4
3
1
cos α = −0,8; tg α = − ; ctg α = −1 .
4
3
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 3. &Oacute;&iuml;&eth;&icirc;&ntilde;&ograve;&egrave;&ograve;&aring;:
&agrave;) (1 − sin α)(1 + sin α);
&aacute;) sin4 α + sin2 α cos2 α + cos2 α;
&acirc;)
tg α sin α
−
.
sin α ctg α
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
&agrave;) (1 − sin α)(1 + sin α) = 1 − sin2 α = cos2 α;
&aacute;) sin4 α + sin2 α cos2 α + cos2 α = sin2 (sin2 α + cos2 α) + cos2 α = sin2 α ⋅ 1 + cos2 α =
= sin2 α + cos2 α = 1;
cos2 α
tg α sin α tg α ctg α − sin α sin α
1 − sin2 α
&acirc;)
=
= cos α;
−
=
=
cos α
cos α
sin α ctg α
sin α ctg α
sin α ⋅
sin α
&Auml;&icirc;&ecirc;&agrave;&aelig;&egrave;&ograve;&aring; &ograve;&icirc;&aelig;&auml;&aring;&ntilde;&ograve;&acirc;&icirc;:
(sin α + cos α) − 1
= 2 tg2 α.
ctg α − sin α cos α
&Auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc;.
&Iuml;&eth;&aring;&icirc;&aacute;&eth;&agrave;&ccedil;&oacute;&aring;&igrave; &euml;&aring;&acirc;&oacute;&thorn; &divide;&agrave;&ntilde;&ograve;&uuml; &ograve;&icirc;&aelig;&auml;&aring;&ntilde;&ograve;&acirc;&agrave;. &Aring;&ntilde;&euml;&egrave; &acirc; &eth;&aring;&ccedil;&oacute;&euml;&uuml;&ograve;&agrave;&ograve;&aring; &iuml;&icirc;&euml;&oacute;&divide;&egrave;&igrave; &iuml;&eth;&agrave;&acirc;&oacute;&thorn; &divide;&agrave;&ntilde;&ograve;&uuml;, &ograve;&icirc; &ograve;&icirc;&aelig;&auml;&aring;&ntilde;&ograve;&acirc;&icirc;
&auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&iacute;&icirc;.
(sin α + cos α)2 − 1 sin2 α + 2 sin α cos α + cos2 α − 1
=
=
cos α
ctg α − sin α cos α
− sin α cos α
sin α
2
=
1 + 2 sin α cos α − 1
cos α − sin2 α cos α
sin α
=
2 sin α cos α ⋅ sin α
cos α(1 − sin2 α)
=
2 cos α ⋅ sin2 α
cos α ⋅ cos2 α
= 2⋅
sin2 α
cos2 α
= 2 tg2 α.
&Ograve;&icirc;&aelig;&auml;&aring;&ntilde;&ograve;&acirc;&icirc; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&iacute;&icirc;.
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 5. &Auml;&agrave;&iacute;&icirc;: sin α + cos α = a.
&Iacute;&agrave;&eacute;&auml;&egrave;&ograve;&aring;: sin αcos α.
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
&Acirc;&icirc;&ccedil;&acirc;&aring;&auml;&aring;&igrave; &icirc;&aacute;&aring; &divide;&agrave;&ntilde;&ograve;&egrave; &eth;&agrave;&acirc;&aring;&iacute;&ntilde;&ograve;&acirc;&agrave; &acirc; &ecirc;&acirc;&agrave;&auml;&eth;&agrave;&ograve;: (sin α + cos α)2 = a2; sin2 α + 2 sin α cos α + cos2 α = a2;
1 + 2 sin α cos α = a2;
a2 − 1
.
2
a2 − 1
&Icirc;&ograve;&acirc;&aring;&ograve;:
.
2
sin α cos α =
34
Раздел 1. Выражения и преобразования
Произведение тангенса и котангенса одного и того же аргумента
π
n, n ∈ Z;
2
1
1
tg α =
, ctg α =
.
ctg α
tg α
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 1. &Igrave;&icirc;&atilde;&oacute;&ograve; &euml;&egrave; &aacute;&ucirc;&ograve;&uuml; &ntilde;&iuml;&eth;&agrave;&acirc;&aring;&auml;&euml;&egrave;&acirc;&ucirc; &eth;&agrave;&acirc;&aring;&iacute;&ntilde;&ograve;&acirc;&agrave;:
tg α ⋅ ctg α = 1 &iuml;&eth;&egrave; α ≠
2
5
1
, ctg α = ; &aacute;) tg α = − , ctg α = 2?
5
2
2
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
&Iuml;&icirc;&ntilde;&ecirc;&icirc;&euml;&uuml;&ecirc;&oacute; tg α ⋅ ctg α = 1, &ograve;&icirc;:
2 5
&agrave;) &eth;&agrave;&acirc;&aring;&iacute;&ntilde;&ograve;&acirc;&agrave; &ntilde;&iuml;&eth;&agrave;&acirc;&aring;&auml;&euml;&egrave;&acirc;&ucirc;, &ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; tg α ctg α = ⋅ = 1;
5 2
1
&aacute;) &eth;&agrave;&acirc;&aring;&iacute;&ntilde;&ograve;&acirc;&agrave; &ntilde;&iuml;&eth;&agrave;&acirc;&aring;&auml;&euml;&egrave;&acirc;&ucirc;&igrave;&egrave; &iacute;&aring; &igrave;&icirc;&atilde;&oacute;&ograve; &aacute;&ucirc;&ograve;&uuml;, &ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; tg α ctg α = − ⋅ 2 = −1 ≠ 1.
2
&agrave;) tg α =
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 2. &Oacute;&iuml;&eth;&icirc;&ntilde;&ograve;&egrave;&ograve;&aring;: tg 1&deg; ⋅ tg 3&deg; ⋅ tg 5&deg; ⋅ …. ⋅ tg 87&deg; ⋅ tg 89&deg;.
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
&Ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; tg (90&deg; − α) = ctg α, &ograve;&icirc;
tg 1&deg; ⋅ tg 3&deg; ⋅ tg 5&deg; ⋅ …. ⋅ tg 87&deg; ⋅ tg 89&deg; = tg 1&deg; ⋅ tg 3&deg; ⋅ … ⋅ tg 43&deg; ⋅ tg 45&deg; ⋅ tg 47&deg; ⋅ … ⋅ tg 87&deg; &times;
&times; tg 89&deg; = tg 1&deg; ⋅ tg 3&deg; ⋅ … ⋅ tg 43&deg; ⋅ 1 ⋅ &ntilde;tg 43&deg; ⋅ … ⋅ ctg 3&deg; ⋅ ctg 1&deg; = (tg 1&deg; ⋅ ctg 1&deg;)(tg 3&deg; ⋅ ctg 3&deg;) ⋅ … ⋅ (tg 43&deg; &times;
&times; ctg 43&deg;) = 1 ⋅ 1 ⋅ … ⋅ 1 = 1.
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 3. &Auml;&agrave;&iacute;&icirc;: tg α + ctg α = 2. &Iacute;&agrave;&eacute;&auml;&egrave;&ograve;&aring;: tg2 α + ctg2 α.
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
&Iuml;&icirc;&ntilde;&ecirc;&icirc;&euml;&uuml;&ecirc;&oacute; tg α + ctg α = 2, &ograve;&icirc; (tg α + ctg α)2 = 22 &egrave;&euml;&egrave; tg2 α + 2 tg α ctg α + ctg2 α = 4. &Oacute;&divide;&egrave;&ograve;&ucirc;&acirc;&agrave;&yuml;,
&divide;&ograve;&icirc; tg α ctg α = 1, &egrave;&ccedil; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&iacute;&aring;&atilde;&icirc; &eth;&agrave;&acirc;&aring;&iacute;&ntilde;&ograve;&acirc;&agrave; tg2 α + ctg2 α = 4 − 2 tg α ctg α = 4 − 2 ⋅ 1 = 2.
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 4. &Oacute;&iuml;&eth;&icirc;&ntilde;&ograve;&egrave;&ograve;&aring;:
(tg α + ctg α)2 − (tg α − ctg α)2.
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
(tg α + ctg α)2 − (tg α − ctg α)2 = (tg2 α + 2 tg α ctg α + ctg2 α) − (tg2 α − 2 tg α ctg α + ctg2 α) =
= tg2 α + 2 ⋅ 1 + ctg2 α − tg2 α + 2 ⋅ 1 − ctg2 α = 2 + 2 = 4.
Зависимость между тангенсом и косинусом одного и того же аргумента
1 + tg2 α =
1
cos2 α
&iuml;&eth;&egrave; α ≠
π
+ πn, n ∈ Z.
2
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 1. &Acirc;&ucirc;&divide;&egrave;&ntilde;&euml;&egrave;&ograve;&aring; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&egrave;&yuml; &ograve;&eth;&egrave;&atilde;&icirc;&iacute;&icirc;&igrave;&aring;&ograve;&eth;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&otilde; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&eacute; &divide;&egrave;&ntilde;&euml;&agrave; α, &aring;&ntilde;&euml;&egrave;
π
tg α = 4, 0 &lt; α &lt; .
2
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
π
&Ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; 0 &lt; α &lt; , &ograve;&icirc; α — ∈ I &divide;&aring;&ograve;&acirc;&aring;&eth;&ograve;&egrave;, &iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute; ctg α &gt; 0; 0 &lt; sin α &lt; 1; 0 &lt; cos α &lt; 1.
2
1
1
1
&Egrave;&ccedil; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc; 1 + tg2 α =
&ntilde;&euml;&aring;&auml;&oacute;&aring;&ograve;:
= 1 + 42 ;
= 17;
2
2
cos α
cos α
cos2 α
1
1
; cos α = &plusmn;
.
17
17
&Oacute;&divide;&egrave;&ograve;&ucirc;&acirc;&agrave;&yuml;, &divide;&ograve;&icirc; 0 &lt; cos α &lt; 1,
cos2 α =
cos α =
1
=
17
1
17
=
17
;
17
35
&Oacute;&divide;&egrave;&ograve;&ucirc;&acirc;&agrave;&yuml;, &divide;&ograve;&icirc; 0 &lt; sinα &lt; 1
sin α = 1 − cos2 α = 1 −
1
=
17
16
=
17
4
17
=
4 17
;
17
1
1
ctg α =
= = 0, 25.
tg α 4
&Icirc;&ograve;&acirc;&aring;&ograve;: sin α =
4 17
17
; cos α =
;
17
17
α = 0,25.
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 2. &Oacute;&iuml;&eth;&icirc;&ntilde;&ograve;&egrave;&ograve;&aring;:
(1 + tg2 α) +
1
sin2 α
.
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
(1 + tg2 α) +
1
sin α
2
=
1
+
cos α
2
1
sin α
2
=
sin2 α + cos2 α
sin2 α cos2 α
=
1
sin2 α cos2 α
.
Зависимость между котангенсом и синусом одного и того же аргумента
1
1 + ctg2 α =
α ≠ πn, n ∈ Z.
sin2 α
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 1. &Acirc;&ucirc;&divide;&egrave;&ntilde;&euml;&egrave;&ograve;&aring; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&egrave;&yuml; &ograve;&eth;&egrave;&atilde;&icirc;&iacute;&icirc;&igrave;&aring;&ograve;&eth;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&otilde; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&eacute;, &aring;&ntilde;&euml;&egrave; ctg α = −3, α — &oacute;&atilde;&icirc;&euml;
IV &divide;&aring;&ograve;&acirc;&aring;&eth;&ograve;&egrave;.
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
&Ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; α — &oacute;&atilde;&icirc;&euml; IV &divide;&aring;&ograve;&acirc;&aring;&eth;&ograve;&egrave;, &ograve;&icirc; tg α &lt; 0; −1 &lt; sin α &lt; 0; 0 &lt; cos α &lt; 1.
&Egrave;&ccedil;&acirc;&aring;&ntilde;&ograve;&iacute;&icirc;, &divide;&ograve;&icirc; 1 + ctg2 α =
1
sin2 α
= 10; sin2 α =
1
sin2 α
. &Icirc;&ograve;&ntilde;&thorn;&auml;&agrave; 1 + 9 =
1
1
; sin α = &plusmn;
.
10
10
&Iacute;&icirc; −1 &lt; sin α &lt; 0, &iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute;
sin α = −
1
10
=−
;
10
10
cos α = 1 − sin2 α = 1 −
1
=
10
9
=
10
3
10
=
3 10
;
10
1
1
1
tg α =
=
=− .
ctg α −3
3
&Icirc;&ograve;&acirc;&aring;&ograve;: sin α = −
10
3 10
1
; cos α =
; tg α = − .
10
10
3
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 2. &Oacute;&iuml;&eth;&icirc;&ntilde;&ograve;&egrave;&ograve;&aring; &acirc;&ucirc;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml;:
sin2 2α + cos2 2α + ctg2 5α;
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
sin2 2α + cos2 2α + ctg2 5α = 1 + ctg2 5α =
36
Раздел 1. Выражения и преобразования
1
sin2 5α
.
1
sin2 α
;
Другие комбинации соотношений между тригонометрическими
функциями одного и того же аргумента
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 1. &Oacute;&iuml;&eth;&icirc;&ntilde;&ograve;&egrave;&ograve;&aring;:
1 − cos2 α
1 + tg2 α
4
4
&aacute;)
cos
α
−
sin
α
+
1;
&acirc;)
;
.
1 − sin2 α
1 + ctg2 α
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
1 − cos2 α sin2 α
&agrave;)
=
= tg2 α;
1 − sin2 α cos2 α
&aacute;) cos4 α − sin4 α + 1 = (cos2 α)2 − (sin2 α)2 + 1 = (cos2 α − sin2 α)(cos2 α + sin2 α) + 1 =
= cos2 α − sin2 α + 1 = cos2 α + (1 − sin2 α) = cos2 α + cos2 α = 2 cos2 α;
1 + tg2 α
1
1
1
sin2 α sin2 α
&acirc;)
=
:
=
⋅
=
= tg2 α.
2
2
2
2
1
1 + ctg α cos α sin α cos α
cos2 α
&agrave;)
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 2. &Oacute;&iuml;&eth;&icirc;&ntilde;&ograve;&egrave;&ograve;&aring; &acirc;&ucirc;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring;:
1
sin α
1 + cos α
&egrave; &iacute;&agrave;&eacute;&auml;&egrave;&ograve;&aring; &aring;&atilde;&icirc; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&egrave;&aring;, &aring;&ntilde;&euml;&egrave; sin α = .
+
2
1 + cos α
sin α
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
sin α
1 + cos α sin2 α + (1 + cos α)2 sin2 α + 1 + 2 cos α + cos2 α
+
=
=
=
1 + cos α
sin α
(1 + cos α) ⋅ sin α
(1 + cos α) sin α
=
1
2 + 2 cos α
2(1 + cos α)
2
2
1
= 2 : = 4.
=
=
. &Aring;&ntilde;&euml;&egrave; sin α = , &ograve;&icirc;
2
(1 + cos α) sin α (1 + cos α) sin α sin α
sin α
2
1.4.3. Формулы сложения
Синус суммы и разности
&Ntilde;&egrave;&iacute;&oacute;&ntilde; &ntilde;&oacute;&igrave;&igrave;&ucirc; &auml;&acirc;&oacute;&otilde; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&icirc;&acirc; &eth;&agrave;&acirc;&aring;&iacute; &ntilde;&oacute;&igrave;&igrave;&aring; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&aring;&auml;&aring;&iacute;&egrave;&eacute; &ntilde;&egrave;&iacute;&oacute;&ntilde;&agrave; &iuml;&aring;&eth;&acirc;&icirc;&atilde;&icirc; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&agrave; &iacute;&agrave; &ecirc;&icirc;&ntilde;&egrave;&iacute;&oacute;&ntilde;
&acirc;&ograve;&icirc;&eth;&icirc;&atilde;&icirc; &egrave; &ecirc;&icirc;&ntilde;&egrave;&iacute;&oacute;&ntilde;&agrave; &iuml;&aring;&eth;&acirc;&icirc;&atilde;&icirc; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&agrave; &iacute;&agrave; &ntilde;&egrave;&iacute;&oacute;&ntilde; &acirc;&ograve;&icirc;&eth;&icirc;&atilde;&icirc;:
sin (α
α + β)
β = sin α ⋅ cos β + cos α ⋅ sin β.
&Ntilde;&egrave;&iacute;&oacute;&ntilde; &eth;&agrave;&ccedil;&iacute;&icirc;&ntilde;&ograve;&egrave; &auml;&acirc;&oacute;&otilde; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&icirc;&acirc; &eth;&agrave;&acirc;&aring;&iacute; &eth;&agrave;&ccedil;&iacute;&icirc;&ntilde;&ograve;&egrave; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&aring;&auml;&aring;&iacute;&egrave;&eacute; &ntilde;&egrave;&iacute;&oacute;&ntilde;&agrave; &iuml;&aring;&eth;&acirc;&icirc;&atilde;&icirc; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&agrave; &iacute;&agrave;
&ecirc;&icirc;&ntilde;&egrave;&iacute;&oacute;&ntilde; &acirc;&ograve;&icirc;&eth;&icirc;&atilde;&icirc; &egrave; &ecirc;&icirc;&ntilde;&egrave;&iacute;&oacute;&ntilde;&agrave; &iuml;&aring;&eth;&acirc;&icirc;&atilde;&icirc; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&agrave; &iacute;&agrave; &ntilde;&egrave;&iacute;&oacute;&ntilde; &acirc;&ograve;&icirc;&eth;&icirc;&atilde;&icirc;:
sin (α
α − β)
β = sin α ⋅ cos β − cos α ⋅ sin β.
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 1. &Acirc;&ucirc;&divide;&egrave;&ntilde;&euml;&egrave;&ograve;&aring;:
&agrave;) sin 15&deg;; &aacute;) sin 75&deg;.
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
&agrave;) sin 15&deg; = sin (45&deg; − 30&deg;) = sin 45&deg; ⋅ cos 30&deg; − cos 45&deg; ⋅ sin 30&deg; =
2
3
2 1
⋅
−
⋅ =
2
2
2 2
6− 2
;
4
&aacute;) sin 75&deg; = sin (45&deg; + 30&deg;) = sin 45&deg; ⋅ cos 30&deg; + cos 45&deg; ⋅ sin 30&deg; =
2
3
2 1
⋅
+
⋅ =
2
2
2 2
6+ 2
.
4
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 2. &Iacute;&agrave;&eacute;&auml;&egrave;&ograve;&aring; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&egrave;&yuml; &acirc;&ucirc;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&eacute;:
7π
π
π
7π
&agrave;) sin 56&deg; cos 34&deg; + cos 56&deg; sin 34&deg;;
&aacute;) sin
cos
− sin
cos
.
12
12
12
12
.
&agrave;) sin 56&deg; cos 34&deg; + cos 56&deg; sin 34&deg; = sin (56&deg; + 34&deg;) = sin 90&deg; = 1;
&aacute;) sin
7π
π
π
7π
π⎞
6π
π
⎛ 7π
cos
− sin
cos
= sin ⎜
− ⎟ = sin
= sin = 1.
⎝
⎠
12
12
12
12
12 12
12
2
1.4. Синус, косинус, тангенс, котангенс
37
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 3. &Oacute;&iuml;&eth;&icirc;&ntilde;&ograve;&egrave;&ograve;&aring;:
cos α + sin α
.
2
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
cos α + sin α
1
1
π
π
⎛π
⎞
=
cos α +
sin α = sin cos α + cos sin α = sin ⎜ + α ⎟ .
⎝4
⎠
4
4
2
2
2
Косинус суммы и разности
&Ecirc;&icirc;&ntilde;&egrave;&iacute;&oacute;&ntilde; &ntilde;&oacute;&igrave;&igrave;&ucirc; &auml;&acirc;&oacute;&otilde; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&icirc;&acirc; &eth;&agrave;&acirc;&aring;&iacute; &eth;&agrave;&ccedil;&iacute;&icirc;&ntilde;&ograve;&egrave; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&aring;&auml;&aring;&iacute;&egrave;&eacute; &ecirc;&icirc;&ntilde;&egrave;&iacute;&oacute;&ntilde;&icirc;&acirc; &yacute;&ograve;&egrave;&otilde; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&icirc;&acirc;
&egrave; &ntilde;&egrave;&iacute;&oacute;&ntilde;&icirc;&acirc; &yacute;&ograve;&egrave;&otilde; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&icirc;&acirc;:
cos (α
α + β)
β = cos α ⋅ cos β − sin α sin β.
&Ecirc;&icirc;&ntilde;&egrave;&iacute;&oacute;&ntilde; &eth;&agrave;&ccedil;&iacute;&icirc;&ntilde;&ograve;&egrave; &auml;&acirc;&oacute;&otilde; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&icirc;&acirc; &eth;&agrave;&acirc;&aring;&iacute; &ntilde;&oacute;&igrave;&igrave;&aring; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&aring;&auml;&aring;&iacute;&egrave;&eacute; &ecirc;&icirc;&ntilde;&egrave;&iacute;&oacute;&ntilde;&icirc;&acirc; &yacute;&ograve;&egrave;&otilde; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&icirc;&acirc;
&egrave; &ntilde;&egrave;&iacute;&oacute;&ntilde;&icirc;&acirc; &yacute;&ograve;&egrave;&otilde; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&icirc;&acirc;:
cos (α
α − β)
β = cos α ⋅ cos β + sin α sin β.
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 1. &Acirc;&ucirc;&divide;&egrave;&ntilde;&euml;&egrave;&ograve;&aring;:
&agrave;) cos 105&deg;; &aacute;) cos 15&deg;.
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
1 2
3
2
&agrave;) cos 105&deg; = cos (60&deg; + 45&deg;) = cos 60&deg; cos 45&deg; − sin 60&deg; sin 45&deg; = ⋅
−
⋅
=
2 2
2
2
&aacute;) cos 15&deg; = cos (60&deg; − 45&deg;) = cos 60&deg; cos 45&deg; + sin 60&deg; sin 45&deg; =
1 2
3
2
⋅
+
⋅
=
2 2
2
2
2− 6
;
4
2+ 6
.
4
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 2. &Oacute;&iuml;&eth;&icirc;&ntilde;&ograve;&egrave;&ograve;&aring; &acirc;&ucirc;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring;:
cos (α + β) + cos (α − β).
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
cos (α + β) + cos (α − β) = cos α cos β − sin α sin β + cos α cos β + sin α sin β = 2 cos α cos β.
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 3. &Acirc;&ucirc;&divide;&egrave;&ntilde;&euml;&egrave;&ograve;&aring;:
⎛ π 4π ⎞
π
4π
4π
π
5π
π 1
cos ⎜
+
cos
cos
− sin
sin
cos
cos
⎟
15
15
⎝
⎠ =
15
15
15
15
15 =
3 = 2 = 1 = 0, 5.
=
π 1 2
cos 0, 3π sin 0, 2π + sin 0, 3π cos 0, 2π sin(0, 3π + 0, 2π) sin 0, 5π
sin
2
Тангенс суммы и разности
38
tg (α + β) =
π
tg α + tg β
α, β, (α + β) ≠ + πn, n ∈ Z ;
2
1 − tg β tg α
tg (α − β) =
π
tg α − tg β
α, β, (α − β) ≠ + πn, n ∈ Z ;
2
1 + tg β tg α
ctg (α + β) =
ctg α ctg β − 1
ctg β + ctg α
, β, (α + β) ≠ πn, n ∈ Z;
ctg (α − β) =
ctg α ctg β + 1
ctg β − ctg α
, β, (α − β) ≠ πn, n ∈ Z.
Раздел 1. Выражения и преобразования
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 1. &Acirc;&ucirc;&divide;&egrave;&ntilde;&euml;&egrave;&ograve;&aring;:
3π
π
7π
3π
tg
+ tg
tg
− tg
10
20
16
16
&agrave;)
; &aacute;)
.
π
3π
7 π 3π
1 − tg
tg
1 + tg
tg
10 20
16 16
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
π
3π
tg
+ tg
5π
π
⎛ π 3π ⎞
10
20
&agrave;)
= tg
= tg = 1.
= tg ⎜
+
π
3π
10 20 ⎟⎠
20
4
⎝
1 − tg
tg
10 20
7π
3π
tg
− tg
4π
π
⎛ 7 π 3π ⎞
16
16
&aacute;)
= tg ⎜
−
= tg
= tg = 1 .
⎟
7 π 3π
⎝ 16 16 ⎠
16
4
1 + tg
tg
16 16
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 2. &Auml;&icirc;&ecirc;&agrave;&aelig;&egrave;&ograve;&aring; &ograve;&icirc;&aelig;&auml;&aring;&ntilde;&ograve;&acirc;&icirc;:
&agrave;) tg 6α − tg 4α − tg 2α = tg 6α tg 4α tg 2α.
&Auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc;.
tg 6α =
tg 4α + tg 2α
⇒
1 − tg 4α tg 2α
α + tg 2α = tg 6α(1 − tg4α tg 2α).
&Ograve;&icirc;&atilde;&auml;&agrave; &egrave;&ccedil; &auml;&agrave;&iacute;&iacute;&icirc;&atilde;&icirc; &eth;&agrave;&acirc;&aring;&iacute;&ntilde;&ograve;&acirc;&agrave; &egrave;&igrave;&aring;&aring;&igrave;:
tg 4α + tg 2α = tg 6α − tg 6α tg 4α tg 2α = tg 6α(1 − tg4α tg 2α).
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 3. &Acirc;&ucirc;&divide;&egrave;&ntilde;&euml;&egrave;&ograve;&aring; tg 15&deg;.
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
tg 45&deg; − tg 30&deg;
tg 15&deg; = tg (45&deg; − 30&deg;) =
=
1 + tg 45&deg; tg 30&deg;
=
( 3 − 1)( 3 − 1)
( 3 + 1)( 3 − 1)
=
1−
1
3 = 3 −1 =
1
3 +1
1 + 1⋅
3
3−2 3 +1 4−2 3
=
= 2 − 3.
3−1
2
1.4.4. Следствия из формул сложения
Синус двойного аргумента
&Ntilde;&egrave;&iacute;&oacute;&ntilde; &auml;&acirc;&icirc;&eacute;&iacute;&icirc;&atilde;&icirc; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&agrave; &eth;&agrave;&acirc;&aring;&iacute; &oacute;&auml;&acirc;&icirc;&aring;&iacute;&iacute;&icirc;&igrave;&oacute; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&aring;&auml;&aring;&iacute;&egrave;&thorn; &ntilde;&egrave;&iacute;&oacute;&ntilde;&agrave; &egrave; &ecirc;&icirc;&ntilde;&egrave;&iacute;&oacute;&ntilde;&agrave; &auml;&agrave;&iacute;&iacute;&icirc;&atilde;&icirc; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&agrave;:
sin 2α
α = 2 sin α cos α.
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 1. &Acirc;&ucirc;&divide;&egrave;&ntilde;&euml;&egrave;&ograve;&aring;: sin 2α, &aring;&ntilde;&euml;&egrave; sin α = −0,6; 180&deg; &lt; α &lt; 270&deg;.
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;. sin 2α = 2 sin α cos α. &Iacute;&agrave;&eacute;&auml;&aring;&igrave; cos α.
&Ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; 180&deg; &lt; α &lt; 270&deg;, &ograve;&icirc; α — &oacute;&atilde;&icirc;&euml; III &divide;&aring;&ograve;&acirc;&aring;&eth;&ograve;&egrave;, &ograve;. &aring;.
−1 &lt; cos α &lt; 0.
cos α = − 1 − sin2 α = − 1 − (−0, 6)2 = − 1 − 0, 36 = − 0, 64 = −0, 8.
&Egrave;&ograve;&agrave;&ecirc;, sin 2α = 2 ⋅ (−0,6) ⋅ (−0,8) = 0,96.
&Icirc;&ograve;&acirc;&aring;&ograve;: 0,96.
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 2. &Acirc;&ucirc;&divide;&egrave;&ntilde;&euml;&egrave;&ograve;&aring;:
&agrave;) sin 15&deg; cos 15&deg;; &aacute;) (cos 75&deg; − sin 75&deg;)2.
1.4. Синус, косинус, тангенс, котангенс
39
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
&agrave;) sin 15&deg; cos 15&deg; =
1
1
1 1 1
(2 sin 15&deg; cos 15&deg;) = sin 30&deg; = ⋅ = ;
2
2
2 2 4
&aacute;) (cos 75&deg; − sin 75&deg;)2 = cos2 75&deg; − 2 cos 75&deg; sin 75&deg; + sin2 75&deg; = (cos2 75&deg; + sin2 75&deg;) −
1 1
– 2 cos 75&deg; sin 75&deg; = 1 − sin 150&deg; = 1 – sin (180&deg; – 30&deg;) = 1 − sin 30&deg; = 1 − = .
2 2
Косинус двойного аргумента
&Ecirc;&icirc;&ntilde;&egrave;&iacute;&oacute;&ntilde; &auml;&acirc;&icirc;&eacute;&iacute;&icirc;&atilde;&icirc; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&agrave; &eth;&agrave;&acirc;&aring;&iacute; &eth;&agrave;&ccedil;&iacute;&icirc;&ntilde;&ograve;&egrave; &ecirc;&acirc;&agrave;&auml;&eth;&agrave;&ograve;&icirc;&acirc; &ecirc;&icirc;&ntilde;&egrave;&iacute;&oacute;&ntilde;&agrave; &egrave; &ntilde;&egrave;&iacute;&oacute;&ntilde;&agrave; &auml;&agrave;&iacute;&iacute;&icirc;&atilde;&icirc; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&agrave;:
cos 2α
α = cos2 α − sin2 α = 2 cos2 α − 1 = 1 − 2 sin2 α.
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 1. &Auml;&icirc;&ecirc;&agrave;&aelig;&egrave;&ograve;&aring; &ograve;&icirc;&aelig;&auml;&aring;&ntilde;&ograve;&acirc;&icirc;:
cos2 (α + β) + cos2 (α − β) − cos 2α cos 2β = 1.
&Auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc;.
&Iuml;&eth;&aring;&icirc;&aacute;&eth;&agrave;&ccedil;&oacute;&aring;&igrave; &euml;&aring;&acirc;&oacute;&thorn; &divide;&agrave;&ntilde;&ograve;&uuml; &eth;&agrave;&acirc;&aring;&iacute;&ntilde;&ograve;&acirc;&agrave;:
cos2(α + β) + cos2(α − β) − cos 2α cos 2β = (cos α cos β − sin α sin β)2 + (cos α cos β + sin α sin β)2 −
− (cos2 α − sin2 α)(cos2 β − sin2 β) = 2 cos2 α cos2 β + 2 sin2 α sin2 β − cos2 α cos2 β +
+ cos2 α sin2 β + sin2 α cos2 β − sin2 α sin2 β = cos2 α cos2 β + sin2 α sin2 β + cos2 α sin2 β + sin2 α cos2 β =
= cos2 α(cos2 β + sin2 β) + sin2 α(sin2 β + cos2 β) = 1, &divide;&ograve;&icirc; &egrave; &ograve;&eth;&aring;&aacute;&icirc;&acirc;&agrave;&euml;&icirc;&ntilde;&uuml; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&uuml;.
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 2. &Acirc;&ucirc;&divide;&egrave;&ntilde;&euml;&egrave;&ograve;&aring;:
π
π
&agrave;) cos2
− sin2 ; &aacute;) cos4 15&deg; − sin4 15&deg;.
8
8
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
π
π
2
2 π
− sin2 = cos =
;
&agrave;) cos
8
8
4
2
cos4 15&deg; − sin4 15&deg; = (cos2 15&deg;)2 − (sin2 15&deg;)2 = (cos2 15&deg; − sin2 15&deg;)(cos2 15&deg; + sin2 15&deg;) =
3
.
2
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 3. &Oacute;&iuml;&eth;&icirc;&ntilde;&ograve;&egrave;&ograve;&aring;:
= cos 30&deg; ⋅ 1 =
&agrave;) 2 cos2 α − cos 2α;
&aacute;)
1 − cos 2α + sin 2α
.
1 + cos 2α + sin 2α
.
&agrave;) 2 cos2 α − cos 2α = 2 cos2 α − (cos2 α − sin2 α) = 2 cos2 α – cos2 α + sin2 α = cos2 α + sin2 α = 1;
1 − cos 2α + sin 2α cos2 α + sin2 α − cos2 α + sin2 α + 2 sin α cos α
&aacute;)
=
=
1 + cos 2α + sin 2α cos2 α + sin2 α + cos2 α − sin2 α + 2 sin α cos α
=
2 sin2 α + 2 sin α cos α
2 cos2 α + 2 sin α cos α
=
2 sin α(sin α + cos α) sin α
=
= tg α.
2 cos α(cos α + sin α) cos α
Тангенс двойного аргумента
&Ograve;&agrave;&iacute;&atilde;&aring;&iacute;&ntilde; &auml;&acirc;&icirc;&eacute;&iacute;&icirc;&atilde;&icirc; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&agrave; &eth;&agrave;&acirc;&aring;&iacute; &divide;&agrave;&ntilde;&ograve;&iacute;&icirc;&igrave;&oacute; &oacute;&auml;&acirc;&icirc;&aring;&iacute;&iacute;&icirc;&atilde;&icirc; &ograve;&agrave;&iacute;&atilde;&aring;&iacute;&ntilde;&agrave; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&agrave; &egrave; &eth;&agrave;&ccedil;&iacute;&icirc;&ntilde;&ograve;&egrave; &aring;&auml;&egrave;&iacute;&egrave;&ouml;&ucirc; &egrave; &ecirc;&acirc;&agrave;&auml;&eth;&agrave;&ograve;&agrave; &ograve;&agrave;&iacute;&atilde;&aring;&iacute;&ntilde;&agrave; &auml;&agrave;&iacute;&iacute;&icirc;&atilde;&icirc; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&agrave;:
2 tg α
.
tg 2 α =
1 − tg 2 α
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 1. &Acirc;&ucirc;&divide;&egrave;&ntilde;&euml;&egrave;&ograve;&aring;:
π
2 tg
6 tg 75&deg;
8
&agrave;)
.
; &aacute;)
π
1 − tg2 75&deg;
1 − tg2
8
40
Раздел 1. Выражения и преобразования
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
π
2 tg
8
&agrave;)
π
1 − tg
8
6 tg 75&deg;
2
&aacute;)
π
⎛ π⎞
= tg ⎜ 2 ⋅ ⎟ = tg = 1;
⎝ 8⎠
4
1 − tg 75&deg;
2
= 3⋅
2 tg 75&deg;
1 − tg 75&deg;
2
= 3 ⋅ tg 150&deg; = 3tg(180&deg; − 30&deg;) = 3 ⋅ (−
−tg30&deg;) = −3 ⋅
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 2. &Acirc;&ucirc;&divide;&egrave;&ntilde;&euml;&egrave;&ograve;&aring; tg 2α, &aring;&ntilde;&euml;&egrave; tg α =
1
3
= − 3.
1
.
2
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
1
2 = 1: 3 = 4 = 11.
tg 2α =
=
4 3
3
1 − tg2 α 1 − 1
4
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 3. &Oacute;&iuml;&eth;&icirc;&ntilde;&ograve;&egrave;&ograve;&aring;:
1
1
−
.
1 − tg α 1 + tg α
2 tg α
2⋅
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
1
1
1 + tg α − (1 − tg α) 1 + tg α − 1 + tg α
2 tg α
−
=
=
=
= tg 2α.
1 − tg α 1 + tg α
(1 − tg α)(1 + tg α)
1 − tg2 α
1 − tg2 α
1.4.5. Формулы приведения
π
3π
&plusmn; α, π &plusmn; α,
&plusmn; α, 2π &plusmn; α &igrave;&icirc;&atilde;&oacute;&ograve; &aacute;&ucirc;&ograve;&uuml; &acirc;&ucirc;&eth;&agrave;&aelig;&aring;&iacute;&ucirc;
2
2
&divide;&aring;&eth;&aring;&ccedil; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&agrave; α &ntilde; &iuml;&icirc;&igrave;&icirc;&ugrave;&uuml;&thorn; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;, &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&aring; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&thorn;&ograve; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&agrave;&igrave;&egrave; &iuml;&eth;&egrave;&acirc;&aring;&auml;&aring;&iacute;&egrave;&yuml;.
&Ograve;&eth;&egrave;&atilde;&icirc;&iacute;&icirc;&igrave;&aring;&ograve;&eth;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&aring; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&icirc;&acirc;
⎛π⎞
&Auml;&acirc;&agrave; &oacute;&atilde;&euml;&agrave; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&thorn;&ograve;&ntilde;&yuml; &auml;&icirc;&iuml;&icirc;&euml;&iacute;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&ucirc;&igrave;&egrave;, &aring;&ntilde;&euml;&egrave; &egrave;&otilde; &ntilde;&oacute;&igrave;&igrave;&agrave; &eth;&agrave;&acirc;&iacute;&agrave; 90&deg; ⎜ ⎟ , &auml;&euml;&yuml; &iacute;&egrave;&otilde; &ntilde;&iuml;&eth;&agrave;&acirc;&aring;&auml;&euml;&egrave;&acirc;&ucirc;
⎝2⎠
&eth;&agrave;&acirc;&aring;&iacute;&ntilde;&ograve;&acirc;&agrave;:
⎛π
⎞
sin α = cos ⎜ − β ⎟
⎝2
⎠
⎛π
⎞
tg α = ctg ⎜ − β ⎟
2
⎝
⎠
⎛π
⎞
cos α = sin ⎜ − β ⎟
⎝2
⎠
⎛π
⎞
ctg α = tg ⎜ − β ⎟ .
⎝2
⎠
&times;&ograve;&icirc;&aacute;&ucirc; &icirc;&aacute;&euml;&aring;&atilde;&divide;&egrave;&ograve;&uuml; &ccedil;&agrave;&iuml;&icirc;&igrave;&egrave;&iacute;&agrave;&iacute;&egrave;&aring; &ocirc;&icirc;&eth;&igrave;&oacute;&euml; &iuml;&eth;&egrave;&acirc;&aring;&auml;&aring;&iacute;&egrave;&yuml; &auml;&euml;&yuml; &iuml;&eth;&aring;&icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&acirc;&agrave;&iacute;&egrave;&yuml; &acirc;&ucirc;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&eacute; &acirc;&egrave;&auml;&agrave;:
⎛ πn
⎞
sin ⎜
&plusmn; α⎟
⎝ 2
⎠
⎛ πn
⎞
tg ⎜
&plusmn; α⎟
⎝ 2
⎠
⎛ πn
⎞
cos ⎜
&plusmn; α⎟
⎝ 2
⎠
⎛ πn
⎞
ctg ⎜
&plusmn; α ⎟ , n ∈ Z,
⎝ 2
⎠
&oacute;&auml;&icirc;&aacute;&iacute;&icirc; &iuml;&icirc;&euml;&uuml;&ccedil;&icirc;&acirc;&agrave;&ograve;&uuml;&ntilde;&yuml; &ograve;&agrave;&ecirc;&egrave;&igrave;&egrave; &iuml;&eth;&agrave;&acirc;&egrave;&euml;&agrave;&igrave;&egrave;:
&agrave;) &iuml;&aring;&eth;&aring;&auml; &iuml;&eth;&egrave;&acirc;&aring;&auml;&aring;&iacute;&iacute;&icirc;&eacute; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&aring;&eacute; &ntilde;&ograve;&agrave;&acirc;&egrave;&ograve;&ntilde;&yuml; &ograve;&icirc;&ograve; &ccedil;&iacute;&agrave;&ecirc;, &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&eacute; &egrave;&igrave;&aring;&aring;&ograve; &egrave;&ntilde;&otilde;&icirc;&auml;&iacute;&agrave;&yuml; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml;, &aring;&ntilde;&euml;&egrave;
0&lt;α&lt;
π
;
2
1.4. Синус, косинус, тангенс, котангенс
41
&aacute;) &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml; &igrave;&aring;&iacute;&yuml;&aring;&ograve;&ntilde;&yuml; &iacute;&agrave; &laquo;&ecirc;&icirc;&ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&thorn;&raquo;, &aring;&ntilde;&euml;&egrave; &iuml; — &iacute;&aring;&divide;&aring;&ograve;&iacute;&icirc;&aring;; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml; &iacute;&aring; &igrave;&aring;&iacute;&yuml;&aring;&ograve;&ntilde;&yuml;, &aring;&ntilde;&euml;&egrave; &iuml; — &divide;&aring;&ograve;&iacute;&icirc;&aring; (&ecirc;&icirc;&ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml;&igrave;&egrave; &ntilde;&egrave;&iacute;&oacute;&ntilde;&agrave;, &ecirc;&icirc;&ntilde;&egrave;&iacute;&oacute;&ntilde;&agrave;, &ograve;&agrave;&iacute;&atilde;&aring;&iacute;&ntilde;&agrave; &egrave; &ecirc;&icirc;&ograve;&agrave;&iacute;&atilde;&aring;&iacute;&ntilde;&agrave; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&thorn;&ograve;&ntilde;&yuml; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc; &ecirc;&icirc;&ntilde;&egrave;&iacute;&oacute;&ntilde;, &ntilde;&egrave;&iacute;&oacute;&ntilde;, &ecirc;&icirc;&ograve;&agrave;&iacute;&atilde;&aring;&iacute;&ntilde; &egrave; &ograve;&agrave;&iacute;&atilde;&aring;&iacute;&ntilde;).
&Iuml;&eth;&egrave;&igrave;&aring;&eth;&ucirc; &ecirc; &yacute;&ograve;&icirc;&igrave;&oacute; &iuml;&eth;&agrave;&acirc;&egrave;&euml;&oacute; &iuml;&eth;&egrave;&acirc;&aring;&auml;&aring;&iacute;&ucirc; &acirc; &ograve;&agrave;&aacute;&euml;&egrave;&ouml;&aring;.
&Agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&ucirc;
&Ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml;
ϕ=
π
&plusmn;α
2
sin ϕ
cos α
cos ϕ
ϕ=π &plusmn; α
ϕ=
3π
&plusmn;α
2
ϕ = 2π − α
−cos α
−sin α
∓ sin α
∓ sin α
−cos α
&plusmn; sin α
cos α
tg ϕ
∓ctg α
&plusmn;tg α
∓ctg α
−tg α
ctg ϕ
∓tg α
&plusmn;ctg α
∓tg α
−ctg α
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 1. &Iacute;&agrave;&eacute;&auml;&egrave;&ograve;&aring; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&egrave;&yuml;:
&agrave;) sin
8π
5π
; &aacute;) tg
.
3
6
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
8π
2π ⎞
2π
π⎞
π
3
⎛
⎛
;
= sin ⎜ 2π +
= sin
= sin ⎜ π − ⎟ = sin =
⎝
⎝
3
3 ⎟⎠
3
3⎠
3
2
5π
π⎞
π
1
⎛
&aacute;) tg
= tg ⎜ π − ⎟ = − tg = −
;.
⎝
6
6⎠
6
3
&agrave;) sin
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 2. &Oacute;&iuml;&eth;&icirc;&ntilde;&ograve;&egrave;&ograve;&aring;:
⎛π
⎞
⎛ 3π
⎞
ctg ⎜ − α ⎟ − tg (π + α) + sin ⎜
− α⎟
⎝2
⎠
⎝ 2
⎠
.
cos(π + α)
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;.
⎛π
⎞
⎛ 3π
⎞
ctg ⎜ − α ⎟ − tg (π + α) + sin ⎜
− α⎟
⎝2
⎠
⎝ 2
⎠ tg α − tg α − cos α cos α
=
=
= 1.
cos(π + α)
− cos α
cos α
π
&aacute;&aring;&ccedil; &iuml;&icirc;&igrave;&icirc;&ugrave;&egrave; &ograve;&agrave;&aacute;&euml;&egrave;&ouml;.
12
π
3
1 − cos
1−
2− 3
1 − cos α
2 π
6
2
2 α
=
=
=
.
&ETH;&aring;&oslash;&aring;&iacute;&egrave;&aring;. &Iuml;&icirc; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&aring; sin
&egrave;&igrave;&aring;&aring;&igrave;: sin
=
12
2
2
4
2
2
&Iuml; &eth; &egrave; &igrave; &aring; &eth; 1. &Acirc;&ucirc;&divide;&egrave;&ntilde;&euml;&egrave;&ograve;&aring; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&egrave;&aring; sin
&Ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; 0 &lt;
π
π
π
π
&lt; , &ograve;&icirc; 0 &lt; sin
&lt; 1 . &Iuml;&icirc;&euml;&oacute;&divide;&egrave;&igrave;: sin
=
12 2
12
12
2− 3
.
2
&Oacute;&iuml;&eth;&icirc;&ntilde;&ograve;&egrave;&igrave; &icirc;&ograve;&acirc;&aring;&ograve;:
2− 3
=
2
2− 3 ⋅ 2
2⋅ 2
=
6− 2
.
4
42
Раздел 1. Выражения и преобразования
4−2 3
2 2
=
( 3 − 1)2
2 2
=
3 −1
2 2
=
3 −1
2 2
=
6− 2
.
4
```