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受欢迎的三角函数 >

tan(x)= 1/(cos^2(x))

解答

tan (x) = \frac{1}{cos ^{2} ( x )}

解答

x \in R 无解

求解步骤

tan (x) = \frac{1}{cos ^{2} ( x )}

两边减去

\frac{1}{cos ^{2} ( x )}

tan (x) - \frac{1}{cos ^{2} ( x )} = 0

化简

tan (x) - \frac{1}{cos ^{2} ( x )} : \frac{cos ^{2} ( x ) tan ( x ) - 1}{cos ^{2} ( x )}

tan (x) - \frac{1}{cos ^{2} ( x )}

将项转换为分式:

tan (x) = \frac{tan ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

= \frac{tan ( x ) cos ^{2} ( x )}{cos ^{2} ( x )} - \frac{1}{cos ^{2} ( x )}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{tan ( x ) cos ^{2} ( x ) - 1}{cos ^{2} ( x )}

\frac{cos ^{2} ( x ) tan ( x ) - 1}{cos ^{2} ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

cos^{2} (x) tan (x) - 1 = 0

使用三角恒等式改写

- 1 + cos^{2} (x) tan (x)

使用基本三角恒等式:

tan (x) = \frac{sin ( x )}{cos ( x )}

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

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

cos^{2} (x) \frac{sin ( x )}{cos ( x )}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

约分:

cos (x)

= sin (x) cos (x)

= - 1 + sin (x) cos (x)

使用倍角公式:

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

sin (x) cos (x) = \frac{sin ( 2 x )}{2}

= - 1 + \frac{sin ( 2 x )}{2}

- 1 + \frac{sin ( 2 x )}{2} = 0

将

1

到右边

- 1 + \frac{sin ( 2 x )}{2} = 0

两边加上

1

- 1 + \frac{sin ( 2 x )}{2} + 1 = 0 + 1

化简

\frac{sin ( 2 x )}{2} = 1

\frac{sin ( 2 x )}{2} = 1

在两边乘以

2

\frac{sin ( 2 x )}{2} = 1

在两边乘以

2

\frac{2 sin ( 2 x )}{2} = 1 \cdot 2

化简

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

x \in R 无解

作图

流行的例子

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2 csc (θ) = 2 cot (θ) + 3 csc (θ)

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cos(θ)=(2sqrt(2))/3

cos (θ) = \frac{2 2}{3}

sqrt(2)cos(x)-sqrt(2)sin(x)=-sqrt(3)

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8cos(2x)=8cos(x)-9

8 cos (2 x) = 8 cos (x) - 9