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受欢迎的三角函数 >

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

解答

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

解答

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

+1

度数

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

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

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

化简

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

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

将项转换为分式:

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

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

- (1 - sin^{2} (x)) : - 1 + sin^{2} (x)

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

打开括号

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

使用加减运算法则

- (- a) = a, - (a) = - a

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

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

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

- cos^{2} (x) + cos^{2} (x) sin (x) = 0

分解

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

- cos^{2} (x) + cos^{2} (x) sin (x)

因式分解出通项

cos^{2} (x)

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

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

分别求解每个部分

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

cos^{2} (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos^{2} (x) = 0

使用法则

x^{n} = 0 \Rightarrow x = 0

cos (x) = 0

cos (x) = 0

的通解

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

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

sin (x) - 1 = 0

将

1

到右边

sin (x) - 1 = 0

两边加上

1

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

化简

sin (x) = 1

sin (x) = 1

sin (x) = 1

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}

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

合并所有解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

作图

流行的例子

r^2=a^2(sin(x)+cos(x))

r^{2} = a^{2} (sin (x) + cos (x))

cot (x) = 3.2404

cot (2 θ) = \frac{5}{12}

sin((-pi)/6+3x)= 1/(sqrt(2))

sin (\frac{- π}{6} + 3 x) = \frac{1}{2}

cos (2 x) = 0.2