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受欢迎的三角函数 >

30sin^2(x)=30+15cos(x)

解答

30 sin^{2} (x) = 30 + 15 cos (x)

解答

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

+1

度数

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

求解步骤

30 sin^{2} (x) = 30 + 15 cos (x)

两边减去

30 + 15 cos (x)

30 sin^{2} (x) - 30 - 15 cos (x) = 0

使用三角恒等式改写

- 30 - 15 cos (x) + 30 sin^{2} (x)

使用毕达哥拉斯恒等式:

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

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

= - 30 - 15 cos (x) + 30 (1 - cos^{2} (x))

化简

- 30 - 15 cos (x) + 30 (1 - cos^{2} (x)) : - 30 cos^{2} (x) - 15 cos (x)

- 30 - 15 cos (x) + 30 (1 - cos^{2} (x))

乘开

30 (1 - cos^{2} (x)) : 30 - 30 cos^{2} (x)

30 (1 - cos^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 30, b = 1, c = cos^{2} (x)

= 30 \cdot 1 - 30 cos^{2} (x)

数字相乘:

30 \cdot 1 = 30

= 30 - 30 cos^{2} (x)

= - 30 - 15 cos (x) + 30 - 30 cos^{2} (x)

化简

- 30 - 15 cos (x) + 30 - 30 cos^{2} (x) : - 30 cos^{2} (x) - 15 cos (x)

- 30 - 15 cos (x) + 30 - 30 cos^{2} (x)

对同类项分组

= - 15 cos (x) - 30 cos^{2} (x) - 30 + 30

- 30 + 30 = 0

= - 30 cos^{2} (x) - 15 cos (x)

= - 30 cos^{2} (x) - 15 cos (x)

= - 30 cos^{2} (x) - 15 cos (x)

- 15 cos (x) - 30 cos^{2} (x) = 0

用替代法求解

- 15 cos (x) - 30 cos^{2} (x) = 0

令

: cos (x) = u

- 15 u - 30 u^{2} = 0

- 15 u - 30 u^{2} = 0 : u = - \frac{1}{2}, u = 0

- 15 u - 30 u^{2} = 0

改写成标准形式

a x^{2} + b x + c = 0

- 30 u^{2} - 15 u = 0

使用求根公式求解

- 30 u^{2} - 15 u = 0

二次方程求根公式:

若

a = - 30, b = - 15, c = 0

u_{1, 2} = \frac{- ( - 15 ) \pm ( - 15 ) ^{2} - 4 ( - 30 ) \cdot 0}{2 ( - 30 )}

u_{1, 2} = \frac{- ( - 15 ) \pm ( - 15 ) ^{2} - 4 ( - 30 ) \cdot 0}{2 ( - 30 )}

(- 15)^{2} - 4 (- 30) \cdot 0 = 15

(- 15)^{2} - 4 (- 30) \cdot 0

使用法则

- (- a) = a

= (- 15)^{2} + 4 \cdot 30 \cdot 0

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- 15)^{2} = 1 5^{2}

= 1 5^{2} + 4 \cdot 30 \cdot 0

使用法则

0 \cdot a = 0

= 1 5^{2} + 0

1 5^{2} + 0 = 1 5^{2}

= 1 5^{2}

使用根式运算法则:

n a^{n} = a,

假定

a \geq 0

= 15

u_{1, 2} = \frac{- ( - 15 ) \pm 15}{2 ( - 30 )}

将解分隔开

u_{1} = \frac{- ( - 15 ) + 15}{2 ( - 30 )}, u_{2} = \frac{- ( - 15 ) - 15}{2 ( - 30 )}

u = \frac{- ( - 15 ) + 15}{2 ( - 30 )} : - \frac{1}{2}

\frac{- ( - 15 ) + 15}{2 ( - 30 )}

去除括号:

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

= \frac{15 + 15}{- 2 \cdot 30}

数字相加:

15 + 15 = 30

= \frac{30}{- 2 \cdot 30}

数字相乘:

2 \cdot 30 = 60

= \frac{30}{- 60}

使用分式法则:

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

= - \frac{30}{60}

约分:

30

= - \frac{1}{2}

u = \frac{- ( - 15 ) - 15}{2 ( - 30 )} : 0

\frac{- ( - 15 ) - 15}{2 ( - 30 )}

去除括号:

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

= \frac{15 - 15}{- 2 \cdot 30}

数字相减:

15 - 15 = 0

= \frac{0}{- 2 \cdot 30}

数字相乘:

2 \cdot 30 = 60

= \frac{0}{- 60}

使用分式法则:

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

= - \frac{0}{60}

使用法则

\frac{0}{a} = 0, a \neq = 0

= - 0

= 0

二次方程组的解是:

u = - \frac{1}{2}, u = 0

u = cos (x)

代回

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

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

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

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

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

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{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

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

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

合并所有解

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

作图

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