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受欢迎的三角函数 >

证明 cos(3x)=cos^3(x)-3sin^2(x)cos(x)

解答

证明

cos (3 x) = cos^{3} (x) - 3 sin^{2} (x) cos (x)

解答

真

求解步骤

cos (3 x) = cos^{3} (x) - 3 sin^{2} (x) cos (x)

调整左侧

cos (3 x)

使用三角恒等式改写

cos (3 x)

改写为

= cos (2 x + x)

使用角和恒等式:

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

= cos (2 x) cos (x) - sin (2 x) sin (x)

使用倍角公式:

sin (2 x) = 2 sin (x) cos (x)

= cos (2 x) cos (x) - 2 sin (x) cos (x) sin (x)

化简

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

cos (2 x) cos (x) - 2 sin (x) cos (x) sin (x)

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

2 sin (x) cos (x) sin (x)

使用指数法则:

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

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

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

数字相加:

1 + 1 = 2

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

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

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

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

使用倍角公式:

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

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

乘开

(cos^{2} (x) - sin^{2} (x)) cos (x) - 2 sin^{2} (x) cos (x) : cos^{3} (x) - 3 sin^{2} (x) cos (x)

(cos^{2} (x) - sin^{2} (x)) cos (x) - 2 sin^{2} (x) cos (x)

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

乘开

cos (x) (cos^{2} (x) - sin^{2} (x)) : cos^{3} (x) - sin^{2} (x) cos (x)

cos (x) (cos^{2} (x) - sin^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = cos (x), b = cos^{2} (x), c = sin^{2} (x)

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

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

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

cos^{2} (x) cos (x)

使用指数法则:

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

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

= cos^{2 + 1} (x)

数字相加:

2 + 1 = 3

= cos^{3} (x)

= cos^{3} (x) - sin^{2} (x) cos (x)

= cos^{3} (x) - sin^{2} (x) cos (x) - 2 sin^{2} (x) cos (x)

同类项相加:

- sin^{2} (x) cos (x) - 2 sin^{2} (x) cos (x) = - 3 sin^{2} (x) cos (x)

= cos^{3} (x) - 3 sin^{2} (x) cos (x)

= cos^{3} (x) - 3 sin^{2} (x) cos (x)

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 (1+cot^2(x))tan(x)=csc(x)sec(x)

p ro v e (1 + cot^{2} (x)) tan (x) = csc (x) sec (x)

证明 sin(a+b)+sin(a-b)=2sin(a)cos(b)

p ro v e sin (a + b) + sin (a - b) = 2 sin (a) cos (b)

证明 (cos(x))(1+tan(x))^2=sec(x)+2sin(x)

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

证明 (1+tan^2(x))/(2tan(x))=csc(2x)

p ro v e \frac{1 + tan ^{2} ( x )}{2 tan ( x )} = csc (2 x)

证明 cos(3θ)=cos^3(θ)-3cos(θ)sin^2(θ)

p ro v e cos (3 θ) = cos^{3} (θ) - 3 cos (θ) sin^{2} (θ)