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受欢迎的三角函数 >

证明 cos(3t)=4cos^3(t)-3cos(t)

解答

证明

cos (3 t) = 4 cos^{3} (t) - 3 cos (t)

解答

真

求解步骤

cos (3 t) = 4 cos^{3} (t) - 3 cos (t)

调整左侧

cos (3 t)

使用三角恒等式改写

cos (3 t)

改写为

= cos (2 t + t)

使用角和恒等式:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (2 t) cos (t) - sin (2 t) sin (t)

使用倍角公式:

sin (2 t) = 2 sin (t) cos (t)

= cos (2 t) cos (t) - 2 sin (t) cos (t) sin (t)

化简

cos (2 t) cos (t) - 2 sin (t) cos (t) sin (t) : cos (t) cos (2 t) - 2 sin^{2} (t) cos (t)

cos (2 t) cos (t) - 2 sin (t) cos (t) sin (t)

2 sin (t) cos (t) sin (t) = 2 sin^{2} (t) cos (t)

2 sin (t) cos (t) sin (t)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

sin (t) sin (t) = sin^{1 + 1} (t)

= 2 cos (t) sin^{1 + 1} (t)

数字相加:

1 + 1 = 2

= 2 cos (t) sin^{2} (t)

= cos (t) cos (2 t) - 2 sin^{2} (t) cos (t)

= cos (t) cos (2 t) - 2 sin^{2} (t) cos (t)

= cos (t) cos (2 t) - 2 sin^{2} (t) cos (t)

使用倍角公式:

cos (2 t) = 2 cos^{2} (t) - 1

= (2 cos^{2} (t) - 1) cos (t) - 2 sin^{2} (t) cos (t)

使用毕达哥拉斯恒等式:

cos^{2} (t) + sin^{2} (t) = 1

sin^{2} (t) = 1 - cos^{2} (t)

= (2 cos^{2} (t) - 1) cos (t) - 2 (1 - cos^{2} (t)) cos (t)

乘开

(2 cos^{2} (t) - 1) cos (t) - 2 (1 - cos^{2} (t)) cos (t) : 4 cos^{3} (t) - 3 cos (t)

(2 cos^{2} (t) - 1) cos (t) - 2 (1 - cos^{2} (t)) cos (t)

= cos (t) (2 cos^{2} (t) - 1) - 2 cos (t) (1 - cos^{2} (t))

乘开

cos (t) (2 cos^{2} (t) - 1) : 2 cos^{3} (t) - cos (t)

cos (t) (2 cos^{2} (t) - 1)

使用分配律:

a (b - c) = ab - a c

a = cos (t), b = 2 cos^{2} (t), c = 1

= cos (t) 2 cos^{2} (t) - cos (t) 1

= 2 cos^{2} (t) cos (t) - 1 cos (t)

化简

2 cos^{2} (t) cos (t) - 1 \cdot cos (t) : 2 cos^{3} (t) - cos (t)

2 cos^{2} (t) cos (t) - 1 cos (t)

2 cos^{2} (t) cos (t) = 2 cos^{3} (t)

2 cos^{2} (t) cos (t)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

cos^{2} (t) cos (t) = cos^{2 + 1} (t)

= 2 cos^{2 + 1} (t)

数字相加:

2 + 1 = 3

= 2 cos^{3} (t)

1 \cdot cos (t) = cos (t)

1 cos (t)

乘以:

1 \cdot cos (t) = cos (t)

= cos (t)

= 2 cos^{3} (t) - cos (t)

= 2 cos^{3} (t) - cos (t)

= 2 cos^{3} (t) - cos (t) - 2 (1 - cos^{2} (t)) cos (t)

乘开

- 2 cos (t) (1 - cos^{2} (t)) : - 2 cos (t) + 2 cos^{3} (t)

- 2 cos (t) (1 - cos^{2} (t))

使用分配律:

a (b - c) = ab - a c

a = - 2 cos (t), b = 1, c = cos^{2} (t)

= - 2 cos (t) 1 - (- 2 cos (t)) cos^{2} (t)

使用加减运算法则

- (- a) = a

= - 2 \cdot 1 cos (t) + 2 cos^{2} (t) cos (t)

化简

- 2 \cdot 1 \cdot cos (t) + 2 cos^{2} (t) cos (t) : - 2 cos (t) + 2 cos^{3} (t)

- 2 \cdot 1 cos (t) + 2 cos^{2} (t) cos (t)

2 \cdot 1 \cdot cos (t) = 2 cos (t)

2 \cdot 1 cos (t)

数字相乘:

2 \cdot 1 = 2

= 2 cos (t)

2 cos^{2} (t) cos (t) = 2 cos^{3} (t)

2 cos^{2} (t) cos (t)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

cos^{2} (t) cos (t) = cos^{2 + 1} (t)

= 2 cos^{2 + 1} (t)

数字相加:

2 + 1 = 3

= 2 cos^{3} (t)

= - 2 cos (t) + 2 cos^{3} (t)

= - 2 cos (t) + 2 cos^{3} (t)

= 2 cos^{3} (t) - cos (t) - 2 cos (t) + 2 cos^{3} (t)

化简

2 cos^{3} (t) - cos (t) - 2 cos (t) + 2 cos^{3} (t) : 4 cos^{3} (t) - 3 cos (t)

2 cos^{3} (t) - cos (t) - 2 cos (t) + 2 cos^{3} (t)

对同类项分组

= 2 cos^{3} (t) + 2 cos^{3} (t) - cos (t) - 2 cos (t)

同类项相加:

2 cos^{3} (t) + 2 cos^{3} (t) = 4 cos^{3} (t)

= 4 cos^{3} (t) - cos (t) - 2 cos (t)

同类项相加:

- cos (t) - 2 cos (t) = - 3 cos (t)

= 4 cos^{3} (t) - 3 cos (t)

= 4 cos^{3} (t) - 3 cos (t)

= 4 cos^{3} (t) - 3 cos (t)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

流行的例子

证明 sec(2x)=(sec^2(x)+sec^4(x))/(2+sec^2(x)-sec^4(x))

p ro v e sec (2 x) = \frac{sec ^{2} ( x ) + sec ^{4} ( x )}{2 + sec ^{2} ( x ) - sec ^{4} ( x )}

证明 1/(csc(x))= 1/(sin(x))

p ro v e \frac{1}{csc ( x )} = \frac{1}{sin ( x )}

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p ro v e 1 + tan^{2} (A) = sec^{3} (A) cos (A)

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p ro v e sec (0) - cos (0) = tan (0) sin (0)

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p ro v e 1 - sin (θ) cos (θ) tan (θ) = cos^{2} (θ)