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受欢迎的三角函数 >

tan(x)= 3/(10cos(x)),0<x<90

解答

tan (x) = \frac{3}{10 cos ( x )}, 0^{\circ} < x < 9 0^{\circ}

解答

x = 0.30469 \dots

求解步骤

tan (x) = \frac{3}{10 cos ( x )}, 0^{\circ} < x < 9 0^{\circ}

两边减去

\frac{3}{10 cos ( x )}

tan (x) - \frac{3}{10 cos ( x )} = 0

化简

tan (x) - \frac{3}{10 cos ( x )} : \frac{10 tan ( x ) cos ( x ) - 3}{10 cos ( x )}

tan (x) - \frac{3}{10 cos ( x )}

将项转换为分式:

tan (x) = \frac{tan ( x ) 10 cos ( x )}{10 cos ( x )}

= \frac{tan ( x ) \cdot 10 cos ( x )}{10 cos ( x )} - \frac{3}{10 cos ( 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 ) \cdot 10 cos ( x ) - 3}{10 cos ( x )}

\frac{10 tan ( x ) cos ( x ) - 3}{10 cos ( x )} = 0

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

10 tan (x) cos (x) - 3 = 0

使用三角恒等式改写

10 tan (x) cos (x) - 3

cos (x) tan (x) = sin (x)

cos (x) tan (x)

使用基本三角恒等式:

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

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

化简

cos (x) \frac{sin ( x )}{cos ( x )} : sin (x)

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

分式相乘:

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

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

约分:

cos (x)

= sin (x)

= sin (x)

= - 3 + 10 sin (x)

- 3 + 10 sin (x) = 0

将

3

到右边

- 3 + 10 sin (x) = 0

两边加上

3

- 3 + 10 sin (x) + 3 = 0 + 3

化简

10 sin (x) = 3

10 sin (x) = 3

两边除以

10

10 sin (x) = 3

两边除以

10

\frac{10 sin ( x )}{10} = \frac{3}{10}

化简

sin (x) = \frac{3}{10}

sin (x) = \frac{3}{10}

使用反三角函数性质

sin (x) = \frac{3}{10}

sin (x) = \frac{3}{10}

的通解

sin (x) = a \Rightarrow x = arcsin (a) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (a) + 36 0^{\circ} n

x = arcsin (\frac{3}{10}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{3}{10}) + 36 0^{\circ} n

x = arcsin (\frac{3}{10}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{3}{10}) + 36 0^{\circ} n

在

0 < x < 9 0^{\circ}

范围内的解

x = arcsin (\frac{3}{10})

以小数形式表示解

x = 0.30469 \dots

作图

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