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受欢迎的三角函数 >

4cos(x)=sin^2(x)-2-3cos^2(x)

解答

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

解答

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

+1

度数

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

求解步骤

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

两边减去

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

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

使用三角恒等式改写

2 - sin^{2} (x) + 3 cos^{2} (x) + 4 cos (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)

= 2 - (1 - cos^{2} (x)) + 3 cos^{2} (x) + 4 cos (x)

化简

2 - (1 - cos^{2} (x)) + 3 cos^{2} (x) + 4 cos (x) : 4 cos^{2} (x) + 4 cos (x) + 1

2 - (1 - cos^{2} (x)) + 3 cos^{2} (x) + 4 cos (x)

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

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

打开括号

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

使用加减运算法则

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

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

= 2 - 1 + cos^{2} (x) + 3 cos^{2} (x) + 4 cos (x)

化简

2 - 1 + cos^{2} (x) + 3 cos^{2} (x) + 4 cos (x) : 4 cos^{2} (x) + 4 cos (x) + 1

2 - 1 + cos^{2} (x) + 3 cos^{2} (x) + 4 cos (x)

同类项相加:

cos^{2} (x) + 3 cos^{2} (x) = 4 cos^{2} (x)

= 2 - 1 + 4 cos^{2} (x) + 4 cos (x)

数字相减:

2 - 1 = 1

= 4 cos^{2} (x) + 4 cos (x) + 1

= 4 cos^{2} (x) + 4 cos (x) + 1

= 4 cos^{2} (x) + 4 cos (x) + 1

1 + 4 cos (x) + 4 cos^{2} (x) = 0

用替代法求解

1 + 4 cos (x) + 4 cos^{2} (x) = 0

令

: cos (x) = u

1 + 4 u + 4 u^{2} = 0

1 + 4 u + 4 u^{2} = 0 : u = - \frac{1}{2}

1 + 4 u + 4 u^{2} = 0

改写成标准形式

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

4 u^{2} + 4 u + 1 = 0

使用求根公式求解

4 u^{2} + 4 u + 1 = 0

二次方程求根公式:

若

a = 4, b = 4, c = 1

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 \cdot 4 \cdot 1}{2 \cdot 4}

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 \cdot 4 \cdot 1}{2 \cdot 4}

4^{2} - 4 \cdot 4 \cdot 1 = 0

4^{2} - 4 \cdot 4 \cdot 1

数字相乘:

4 \cdot 4 \cdot 1 = 16

= 4^{2} - 16

4^{2} = 16

= 16 - 16

数字相减:

16 - 16 = 0

= 0

u_{1, 2} = \frac{- 4 \pm 0}{2 \cdot 4}

u = \frac{- 4}{2 \cdot 4}

\frac{- 4}{2 \cdot 4} = - \frac{1}{2}

\frac{- 4}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{- 4}{8}

使用分式法则:

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

= - \frac{4}{8}

约分:

4

= - \frac{1}{2}

u = - \frac{1}{2}

二次方程组的解是:

u = - \frac{1}{2}

u = cos (x)

代回

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

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

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

合并所有解

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

作图

流行的例子

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50 sin (x) + 15 cos (x) = 40, 0 < x < π

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sin (x) = - 0.2761

sin (2 y) = 0