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受欢迎的三角函数 >

48sin^2(x)=48-24cos(x)

解答

48 sin^{2} (x) = 48 - 24 cos (x)

解答

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

+1

度数

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

求解步骤

48 sin^{2} (x) = 48 - 24 cos (x)

两边减去

48 - 24 cos (x)

48 sin^{2} (x) - 48 + 24 cos (x) = 0

使用三角恒等式改写

- 48 + 24 cos (x) + 48 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)

= - 48 + 24 cos (x) + 48 (1 - cos^{2} (x))

化简

- 48 + 24 cos (x) + 48 (1 - cos^{2} (x)) : 24 cos (x) - 48 cos^{2} (x)

- 48 + 24 cos (x) + 48 (1 - cos^{2} (x))

乘开

48 (1 - cos^{2} (x)) : 48 - 48 cos^{2} (x)

48 (1 - cos^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 48, b = 1, c = cos^{2} (x)

= 48 \cdot 1 - 48 cos^{2} (x)

数字相乘:

48 \cdot 1 = 48

= 48 - 48 cos^{2} (x)

= - 48 + 24 cos (x) + 48 - 48 cos^{2} (x)

化简

- 48 + 24 cos (x) + 48 - 48 cos^{2} (x) : 24 cos (x) - 48 cos^{2} (x)

- 48 + 24 cos (x) + 48 - 48 cos^{2} (x)

对同类项分组

= 24 cos (x) - 48 cos^{2} (x) - 48 + 48

- 48 + 48 = 0

= 24 cos (x) - 48 cos^{2} (x)

= 24 cos (x) - 48 cos^{2} (x)

= 24 cos (x) - 48 cos^{2} (x)

24 cos (x) - 48 cos^{2} (x) = 0

用替代法求解

24 cos (x) - 48 cos^{2} (x) = 0

令

: cos (x) = u

24 u - 48 u^{2} = 0

24 u - 48 u^{2} = 0 : u = 0, u = \frac{1}{2}

24 u - 48 u^{2} = 0

改写成标准形式

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

- 48 u^{2} + 24 u = 0

使用求根公式求解

- 48 u^{2} + 24 u = 0

二次方程求根公式:

若

a = - 48, b = 24, c = 0

u_{1, 2} = \frac{- 24 \pm 2 4 ^{2} - 4 ( - 48 ) \cdot 0}{2 ( - 48 )}

u_{1, 2} = \frac{- 24 \pm 2 4 ^{2} - 4 ( - 48 ) \cdot 0}{2 ( - 48 )}

2 4^{2} - 4 (- 48) \cdot 0 = 24

2 4^{2} - 4 (- 48) \cdot 0

使用法则

- (- a) = a

= 2 4^{2} + 4 \cdot 48 \cdot 0

使用法则

0 \cdot a = 0

= 2 4^{2} + 0

2 4^{2} + 0 = 2 4^{2}

= 2 4^{2}

使用根式运算法则:

n a^{n} = a,

假定

a \geq 0

= 24

u_{1, 2} = \frac{- 24 \pm 24}{2 ( - 48 )}

将解分隔开

u_{1} = \frac{- 24 + 24}{2 ( - 48 )}, u_{2} = \frac{- 24 - 24}{2 ( - 48 )}

u = \frac{- 24 + 24}{2 ( - 48 )} : 0

\frac{- 24 + 24}{2 ( - 48 )}

去除括号:

(- a) = - a

= \frac{- 24 + 24}{- 2 \cdot 48}

数字相加/相减:

- 24 + 24 = 0

= \frac{0}{- 2 \cdot 48}

数字相乘:

2 \cdot 48 = 96

= \frac{0}{- 96}

使用分式法则:

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

= - \frac{0}{96}

使用法则

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

= - 0

= 0

u = \frac{- 24 - 24}{2 ( - 48 )} : \frac{1}{2}

\frac{- 24 - 24}{2 ( - 48 )}

去除括号:

(- a) = - a

= \frac{- 24 - 24}{- 2 \cdot 48}

数字相减:

- 24 - 24 = - 48

= \frac{- 48}{- 2 \cdot 48}

数字相乘:

2 \cdot 48 = 96

= \frac{- 48}{- 96}

使用分式法则:

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

= \frac{48}{96}

约分:

48

= \frac{1}{2}

二次方程组的解是:

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

u = cos (x)

代回

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

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

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

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 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{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

合并所有解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

作图

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