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受欢迎的三角函数 >

3cos^2(x)=sin^2(x)

解答

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

解答

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

+1

度数

x = 12 0^{\circ} + 18 0^{\circ} n, x = 6 0^{\circ} + 18 0^{\circ} n

求解步骤

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

两边减去

sin^{2} (x)

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

分解

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

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

将

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

改写为

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

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

使用根式运算法则:

a = (a)^{2}

3 = (3)^{2}

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

使用指数法则:

a^{m} b^{m} = (ab)^{m}

(3)^{2} cos^{2} (x) = (3 cos (x))^{2}

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

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

使用平方差公式:

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

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

= (3 cos (x) + sin (x)) (3 cos (x) - sin (x))

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

分别求解每个部分

3 cos (x) + sin (x) = 0 or 3 cos (x) - sin (x) = 0

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

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

使用三角恒等式改写

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

在两边除以

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

\frac{3 cos ( x ) + sin ( x )}{cos ( x )} = \frac{0}{cos ( x )}

化简

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

使用基本三角恒等式:

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

3 + tan (x) = 0

3 + tan (x) = 0

将

3

到右边

3 + tan (x) = 0

两边减去

3

3 + tan (x) - 3 = 0 - 3

化简

tan (x) = - 3

tan (x) = - 3

tan (x) = - 3

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{2 π}{3} + πn

x = \frac{2 π}{3} + πn

3 cos (x) - sin (x) = 0 : x = \frac{π}{3} + πn

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

使用三角恒等式改写

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

在两边除以

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

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

化简

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

使用基本三角恒等式:

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

3 - tan (x) = 0

3 - tan (x) = 0

将

3

到右边

3 - tan (x) = 0

两边减去

3

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

化简

- tan (x) = - 3

- tan (x) = - 3

两边除以

- 1

- tan (x) = - 3

两边除以

- 1

\frac{- tan ( x )}{- 1} = \frac{- 3}{- 1}

化简

tan (x) = 3

tan (x) = 3

tan (x) = 3

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}{3} + πn

x = \frac{π}{3} + πn

合并所有解

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

作图

流行的例子

sin(2x)+sin(x)=0,0<= x<= 2pi

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

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4sec(x)+4tan(x)=4

4 sec (x) + 4 tan (x) = 4