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受欢迎的三角函数 >

7sin^2(x)+2sin^2(x)-3cos^2(x)=0

解答

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

解答

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

+1

度数

x = 15 0^{\circ} + 18 0^{\circ} n, x = 3 0^{\circ} + 18 0^{\circ} n

求解步骤

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

分解

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

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

同类项相加:

7 sin^{2} (x) + 2 sin^{2} (x) = 9 sin^{2} (x)

= 9 sin^{2} (x) - 3 cos^{2} (x)

将

9

改写为

3 \cdot 3

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

因式分解出通项

3

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

分解

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

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

将

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

改写为

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

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

使用根式运算法则:

a = (a)^{2}

3 = (3)^{2}

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

使用指数法则:

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

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

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

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

使用平方差公式:

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

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

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

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

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

分别求解每个部分

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

3 sin (x) + cos (x) = 0 : x = \frac{5 π}{6} + πn

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

使用三角恒等式改写

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

在两边除以

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

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

化简

\frac{3 sin ( x )}{cos ( x )} + 1 = 0

使用基本三角恒等式:

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

3 tan (x) + 1 = 0

3 tan (x) + 1 = 0

将

1

到右边

3 tan (x) + 1 = 0

两边减去

1

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

化简

3 tan (x) = - 1

3 tan (x) = - 1

两边除以

3

3 tan (x) = - 1

两边除以

3

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

化简

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

化简

\frac{3 tan ( x )}{3} : tan (x)

\frac{3 tan ( x )}{3}

约分:

3

= tan (x)

化简

\frac{- 1}{3} : - \frac{3}{3}

\frac{- 1}{3}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1}{3}

- \frac{1}{3}

有理化:

- \frac{3}{3}

- \frac{1}{3}

乘以共轭根式

\frac{3}{3}

= - \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

使用根式运算法则:

a a = a

33 = 3

= 3

= - \frac{3}{3}

= - \frac{3}{3}

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

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

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

tan (x) = - \frac{3}{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{5 π}{6} + πn

x = \frac{5 π}{6} + πn

3 sin (x) - cos (x) = 0 : x = \frac{π}{6} + πn

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

使用三角恒等式改写

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

在两边除以

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

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

化简

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

使用基本三角恒等式:

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

3 tan (x) - 1 = 0

3 tan (x) - 1 = 0

将

1

到右边

3 tan (x) - 1 = 0

两边加上

1

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

化简

3 tan (x) = 1

3 tan (x) = 1

两边除以

3

3 tan (x) = 1

两边除以

3

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

化简

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

化简

\frac{3 tan ( x )}{3} : tan (x)

\frac{3 tan ( x )}{3}

约分:

3

= tan (x)

化简

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

乘以共轭根式

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

使用根式运算法则:

a a = a

33 = 3

= 3

= \frac{3}{3}

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

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

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

tan (x) = \frac{3}{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}{6} + πn

x = \frac{π}{6} + πn

合并所有解

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

作图

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