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受欢迎的三角函数 >

证明 3sin^2(x)+4cos^2(x)=3+cos^2(x)

解答

证明

3 sin^{2} (x) + 4 cos^{2} (x) = 3 + cos^{2} (x)

解答

真

求解步骤

3 sin^{2} (x) + 4 cos^{2} (x) = 3 + cos^{2} (x)

调整左侧

3 sin^{2} (x) + 4 cos^{2} (x)

使用三角恒等式改写

3 sin^{2} (x) + 4 cos^{2} (x)

使用毕达哥拉斯恒等式:

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

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

= 3 (1 - cos^{2} (x)) + 4 cos^{2} (x)

化简

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

3 (1 - cos^{2} (x)) + 4 cos^{2} (x)

乘开

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

3 (1 - cos^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 3, b = 1, c = cos^{2} (x)

= 3 \cdot 1 - 3 cos^{2} (x)

数字相乘:

3 \cdot 1 = 3

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

= 3 - 3 cos^{2} (x) + 4 cos^{2} (x)

同类项相加:

- 3 cos^{2} (x) + 4 cos^{2} (x) = cos^{2} (x)

= 3 + cos^{2} (x)

= 3 + cos^{2} (x)

= 3 + cos^{2} (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 tan(pi/2-θ)tan(θ)=1

p ro v e tan (\frac{π}{2} - θ) tan (θ) = 1

证明 (1-sin^2(x))csc^2(x)=cot^2(x)

p ro v e (1 - sin^{2} (x)) csc^{2} (x) = cot^{2} (x)

证明 sin(t)tan(t)=(1-cos^2(t))/(cos(t))

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

证明 (csc(x))/(sin(x))-(cot(x))/(tan(x))=1

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

证明 (csc(x)-cot(x))/(sec(x)-1)=cot(x)

p ro v e \frac{csc ( x ) - cot ( x )}{sec ( x ) - 1} = cot (x)