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受欢迎的三角函数 >

sec(x)=2csc(x)

解答

sec (x) = 2 csc (x)

解答

x = 1.10714 \dots + πn

+1

度数

x = 63.43494 \dots^{\circ} + 18 0^{\circ} n

求解步骤

sec (x) = 2 csc (x)

两边减去

2 csc (x)

sec (x) - 2 csc (x) = 0

用 sin, cos 表示

sec (x) - 2 csc (x)

使用基本三角恒等式:

sec (x) = \frac{1}{cos ( x )}

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

使用基本三角恒等式:

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

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

化简

\frac{1}{cos ( x )} - 2 \cdot \frac{1}{sin ( x )} : \frac{sin ( x ) - 2 cos ( x )}{cos ( x ) sin ( x )}

\frac{1}{cos ( x )} - 2 \cdot \frac{1}{sin ( x )}

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

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

分式相乘:

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

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

数字相乘:

1 \cdot 2 = 2

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

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

cos (x), sin (x)

的最小公倍数:

cos (x) sin (x)

cos (x), sin (x)

最小公倍数 (LCM)

计算出由出现在

cos (x)

或

sin (x)

中的因子组成的表达式

= cos (x) sin (x)

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

cos (x) sin (x)

对于

\frac{1}{cos ( x )} :

将分母和分子乘以

sin (x)

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

对于

\frac{2}{sin ( x )} :

将分母和分子乘以

cos (x)

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

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

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

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

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

sin (x) - 2 cos (x) = 0

使用三角恒等式改写

sin (x) - 2 cos (x) = 0

在两边除以

cos (x), cos (x) \neq = 0

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

化简

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

使用基本三角恒等式:

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

tan (x) - 2 = 0

tan (x) - 2 = 0

将

2

到右边

tan (x) - 2 = 0

两边加上

2

tan (x) - 2 + 2 = 0 + 2

化简

tan (x) = 2

tan (x) = 2

使用反三角函数性质

tan (x) = 2

tan (x) = 2

的通解

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (2) + πn

x = arctan (2) + πn

以小数形式表示解

x = 1.10714 \dots + πn

作图

流行的例子

cos^{2} (x) = 0.75

tan (x) = \frac{π}{2}

tan (x) = \frac{π}{4}

13cos(x)+7=-cos(x)

13 cos (x) + 7 = - cos (x)

cos^{2} (x) = 0.25