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受欢迎的三角函数 >

sin^3(x)=cos^3(x)

解答

sin^{3} (x) = cos^{3} (x)

解答

x = \frac{π}{4} + πn

+1

度数

x = 4 5^{\circ} + 18 0^{\circ} n

求解步骤

sin^{3} (x) = cos^{3} (x)

两边减去

cos^{3} (x)

sin^{3} (x) - cos^{3} (x) = 0

分解

sin^{3} (x) - cos^{3} (x) : (sin (x) - cos (x)) (sin^{2} (x) + sin (x) cos (x) + cos^{2} (x))

sin^{3} (x) - cos^{3} (x)

使用立方差公式:

x^{3} - y^{3} = (x - y) (x^{2} + x y + y^{2})

sin^{3} (x) - cos^{3} (x) = (sin (x) - cos (x)) (sin^{2} (x) + sin (x) cos (x) + cos^{2} (x))

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

分别求解每个部分

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

- cos (x) + sin (x) = 0 : x = \frac{π}{4} + πn

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

使用三角恒等式改写

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

在两边除以

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

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

化简

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

使用基本三角恒等式:

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

- 1 + tan (x) = 0

- 1 + tan (x) = 0

将

1

到右边

- 1 + tan (x) = 0

两边加上

1

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

化简

tan (x) = 1

tan (x) = 1

tan (x) = 1

的通解

tan (x)

周期表（周期为

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

cos (x) sin (x) + 1 = 0 :

无解

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

使用三角恒等式改写

cos (x) sin (x) + 1

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

两边减去

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} = 2 (- 1)

化简

sin (2 x) = - 2

sin (2 x) = - 2

- 1 \leq sin (x) \leq 1

无解

合并所有解

x = \frac{π}{4} + πn

作图

流行的例子

sec(x)sin(x)+cos(x)=sec(x)

sec (x) sin (x) + cos (x) = sec (x)

sin(x)+4csc(x)+5=0,0<= x<= 2pi

sin (x) + 4 csc (x) + 5 = 0, 0 \leq x \leq 2 π

r = a sin (3 x)

sin (x) = \frac{18}{25}

cos^{2} (t) = 0