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受欢迎的三角函数 >

4sin(x)-sec(x)=0

解答

4 sin (x) - sec (x) = 0

解答

x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

+1

度数

x = 1 5^{\circ} + 18 0^{\circ} n, x = 7 5^{\circ} + 18 0^{\circ} n

求解步骤

4 sin (x) - sec (x) = 0

用 sin, cos 表示

- sec (x) + 4 sin (x)

使用基本三角恒等式:

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

= - \frac{1}{cos ( x )} + 4 sin (x)

化简

- \frac{1}{cos ( x )} + 4 sin (x) : \frac{- 1 + 4 sin ( x ) cos ( x )}{cos ( x )}

- \frac{1}{cos ( x )} + 4 sin (x)

将项转换为分式:

4 sin (x) = \frac{4 sin ( x ) cos ( x )}{cos ( x )}

= - \frac{1}{cos ( x )} + \frac{4 sin ( x ) cos ( 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{- 1 + 4 sin ( x ) cos ( x )}{cos ( x )}

= \frac{- 1 + 4 sin ( x ) cos ( x )}{cos ( x )}

\frac{- 1 + 4 cos ( x ) sin ( x )}{cos ( x )} = 0

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

- 1 + 4 cos (x) sin (x) = 0

使用三角恒等式改写

- 1 + 4 cos (x) sin (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 + 4 \cdot \frac{sin ( 2 x )}{2}

- 1 + 4 \cdot \frac{sin ( 2 x )}{2} = 0

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

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

分式相乘:

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

= \frac{sin ( 2 x ) \cdot 4}{2}

数字相除:

\frac{4}{2} = 2

= 2 sin (2 x)

- 1 + 2 sin (2 x) = 0

将

1

到右边

- 1 + 2 sin (2 x) = 0

两边加上

1

- 1 + 2 sin (2 x) + 1 = 0 + 1

化简

2 sin (2 x) = 1

2 sin (2 x) = 1

两边除以

2

2 sin (2 x) = 1

两边除以

2

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

化简

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

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

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

的通解

sin (x)

周期表（周期为

2 πn

"）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

2 x = \frac{π}{6} + 2 πn, 2 x = \frac{5 π}{6} + 2 πn

2 x = \frac{π}{6} + 2 πn, 2 x = \frac{5 π}{6} + 2 πn

解

2 x = \frac{π}{6} + 2 πn : x = \frac{π}{12} + πn

2 x = \frac{π}{6} + 2 πn

两边除以

2

2 x = \frac{π}{6} + 2 πn

两边除以

2

\frac{2 x}{2} = \frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

化简

\frac{2 x}{2} = \frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

\frac{\frac{π}{6}}{2} + \frac{2 πn}{2} : \frac{π}{12} + πn

\frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{6}}{2} = \frac{π}{12}

\frac{\frac{π}{6}}{2}

使用分式法则:

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

= \frac{π}{6 \cdot 2}

数字相乘:

6 \cdot 2 = 12

= \frac{π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{π}{12} + πn

x = \frac{π}{12} + πn

x = \frac{π}{12} + πn

x = \frac{π}{12} + πn

解

2 x = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{12} + πn

2 x = \frac{5 π}{6} + 2 πn

两边除以

2

2 x = \frac{5 π}{6} + 2 πn

两边除以

2

\frac{2 x}{2} = \frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

化简

\frac{2 x}{2} = \frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2} : \frac{5 π}{12} + πn

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{5 π}{6}}{2} = \frac{5 π}{12}

\frac{\frac{5 π}{6}}{2}

使用分式法则:

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

= \frac{5 π}{6 \cdot 2}

数字相乘:

6 \cdot 2 = 12

= \frac{5 π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

作图

流行的例子

sin^2(x)=(1-cos^2(x))/2

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

cot(θ)=cos(θ)

cot (θ) = cos (θ)

3csc^3(x)-4csc(x)=0

3 csc^{3} (x) - 4 csc (x) = 0

tan(θ)=(6.06)/(-33.5)

tan (θ) = \frac{6.06}{- 33.5}

csc(θ)=(sqrt(13))/2 ,sec(θ)=(sqrt(13))/3

csc (θ) = \frac{13}{2}, sec (θ) = \frac{13}{3}