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受欢迎的三角函数 >

cos(θ)=cos(3θ)

解答

cos (θ) = cos (3 θ)

解答

θ = πn, θ = \frac{π}{2} + πn

+1

度数

θ = 0^{\circ} + 18 0^{\circ} n, θ = 9 0^{\circ} + 18 0^{\circ} n

求解步骤

cos (θ) = cos (3 θ)

两边减去

cos (3 θ)

cos (θ) - cos (3 θ) = 0

使用三角恒等式改写

- cos (3 θ) + cos (θ)

使用和差化积恒等式:

cos (s) - cos (t) = - 2 sin (\frac{s + t}{2}) sin (\frac{s - t}{2})

= - 2 sin (\frac{θ + 3 θ}{2}) sin (\frac{θ - 3 θ}{2})

化简

- 2 sin (\frac{θ + 3 θ}{2}) sin (\frac{θ - 3 θ}{2}) : 2 sin (θ) sin (2 θ)

- 2 sin (\frac{θ + 3 θ}{2}) sin (\frac{θ - 3 θ}{2})

\frac{θ + 3 θ}{2} = 2 θ

\frac{θ + 3 θ}{2}

同类项相加:

θ + 3 θ = 4 θ

= \frac{4 θ}{2}

数字相除:

\frac{4}{2} = 2

= 2 θ

= - 2 sin (2 θ) sin (\frac{θ - 3 θ}{2})

\frac{θ - 3 θ}{2} = - θ

\frac{θ - 3 θ}{2}

同类项相加:

θ - 3 θ = - 2 θ

= \frac{- 2 θ}{2}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{2 θ}{2}

数字相除:

\frac{2}{2} = 1

= - θ

= - 2 sin (2 θ) sin (- θ)

使用负角恒等式:

sin (- x) = - sin (x)

= - 2 (- sin (θ)) sin (2 θ)

使用法则

- (- a) = a

= 2 sin (θ) sin (2 θ)

= 2 sin (θ) sin (2 θ)

2 sin (2 θ) sin (θ) = 0

分别求解每个部分

sin (2 θ) = 0 or sin (θ) = 0

sin (2 θ) = 0 : θ = πn, θ = \frac{π}{2} + πn

sin (2 θ) = 0

sin (2 θ) = 0

的通解

sin (x)

周期表（周期为

2 πn

"）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

2 θ = 0 + 2 πn, 2 θ = π + 2 πn

2 θ = 0 + 2 πn, 2 θ = π + 2 πn

解

2 θ = 0 + 2 πn : θ = πn

2 θ = 0 + 2 πn

0 + 2 πn = 2 πn

2 θ = 2 πn

两边除以

2

2 θ = 2 πn

两边除以

2

\frac{2 θ}{2} = \frac{2 πn}{2}

化简

θ = πn

θ = πn

解

2 θ = π + 2 πn : θ = \frac{π}{2} + πn

2 θ = π + 2 πn

两边除以

2

2 θ = π + 2 πn

两边除以

2

\frac{2 θ}{2} = \frac{π}{2} + \frac{2 πn}{2}

化简

θ = \frac{π}{2} + πn

θ = \frac{π}{2} + πn

θ = πn, θ = \frac{π}{2} + πn

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

sin (θ) = 0

sin (θ) = 0

的通解

sin (x)

周期表（周期为

2 πn

"）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

θ = 0 + 2 πn, θ = π + 2 πn

θ = 0 + 2 πn, θ = π + 2 πn

解

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn, θ = π + 2 πn

合并所有解

θ = πn, θ = \frac{π}{2} + πn, θ = 2 πn, θ = π + 2 πn

合并重叠的区间

θ = πn, θ = \frac{π}{2} + πn

作图

流行的例子

cot(3θ)=sqrt(3)

cot (3 θ) = 3

tan (x) = \frac{45}{35}

(625^{cos^2(x)})/(25^{cos(x))}=1

\frac{62 5 ^{c o s^{2} (x)}}{2 5 ^{c o s (x)}} = 1

2cos(x)-3sec(x)=5

2 cos (x) - 3 sec (x) = 5

cos(2x)=1-2sin(x),0<= x<2pi

cos (2 x) = 1 - 2 sin (x), 0 \leq x < 2 π