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受欢迎的三角函数 >

5sin(x)-4=2cos(2x)

解答

5 sin (x) - 4 = 2 cos (2 x)

解答

x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn

+1

度数

x = 48.59037 \dots^{\circ} + 36 0^{\circ} n, x = 131.40962 \dots^{\circ} + 36 0^{\circ} n

求解步骤

5 sin (x) - 4 = 2 cos (2 x)

两边减去

2 cos (2 x)

5 sin (x) - 4 - 2 cos (2 x) = 0

使用三角恒等式改写

- 4 - 2 cos (2 x) + 5 sin (x)

使用倍角公式:

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

= - 4 - 2 (1 - 2 sin^{2} (x)) + 5 sin (x)

化简

- 4 - 2 (1 - 2 sin^{2} (x)) + 5 sin (x) : 4 sin^{2} (x) + 5 sin (x) - 6

- 4 - 2 (1 - 2 sin^{2} (x)) + 5 sin (x)

乘开

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

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

使用分配律:

a (b - c) = ab - a c

a = - 2, b = 1, c = 2 sin^{2} (x)

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

使用加减运算法则

- (- a) = a

= - 2 \cdot 1 + 2 \cdot 2 sin^{2} (x)

化简

- 2 \cdot 1 + 2 \cdot 2 sin^{2} (x) : - 2 + 4 sin^{2} (x)

- 2 \cdot 1 + 2 \cdot 2 sin^{2} (x)

数字相乘:

2 \cdot 1 = 2

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

数字相乘:

2 \cdot 2 = 4

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

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

= - 4 - 2 + 4 sin^{2} (x) + 5 sin (x)

数字相减:

- 4 - 2 = - 6

= 4 sin^{2} (x) + 5 sin (x) - 6

= 4 sin^{2} (x) + 5 sin (x) - 6

- 6 + 4 sin^{2} (x) + 5 sin (x) = 0

用替代法求解

- 6 + 4 sin^{2} (x) + 5 sin (x) = 0

令

: sin (x) = u

- 6 + 4 u^{2} + 5 u = 0

- 6 + 4 u^{2} + 5 u = 0 : u = \frac{3}{4}, u = - 2

- 6 + 4 u^{2} + 5 u = 0

改写成标准形式

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

4 u^{2} + 5 u - 6 = 0

使用求根公式求解

4 u^{2} + 5 u - 6 = 0

二次方程求根公式:

若

a = 4, b = 5, c = - 6

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 4 ( - 6 )}{2 \cdot 4}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 4 ( - 6 )}{2 \cdot 4}

5^{2} - 4 \cdot 4 (- 6) = 11

5^{2} - 4 \cdot 4 (- 6)

使用法则

- (- a) = a

= 5^{2} + 4 \cdot 4 \cdot 6

数字相乘:

4 \cdot 4 \cdot 6 = 96

= 5^{2} + 96

5^{2} = 25

= 25 + 96

数字相加:

25 + 96 = 121

= 121

因式分解数字:

121 = 1 1^{2}

= 1 1^{2}

使用根式运算法则:

n a^{n} = a

1 1^{2} = 11

= 11

u_{1, 2} = \frac{- 5 \pm 11}{2 \cdot 4}

将解分隔开

u_{1} = \frac{- 5 + 11}{2 \cdot 4}, u_{2} = \frac{- 5 - 11}{2 \cdot 4}

u = \frac{- 5 + 11}{2 \cdot 4} : \frac{3}{4}

\frac{- 5 + 11}{2 \cdot 4}

数字相加/相减:

- 5 + 11 = 6

= \frac{6}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{6}{8}

约分:

2

= \frac{3}{4}

u = \frac{- 5 - 11}{2 \cdot 4} : - 2

\frac{- 5 - 11}{2 \cdot 4}

数字相减:

- 5 - 11 = - 16

= \frac{- 16}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{- 16}{8}

使用分式法则:

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

= - \frac{16}{8}

数字相除:

\frac{16}{8} = 2

= - 2

二次方程组的解是:

u = \frac{3}{4}, u = - 2

u = sin (x)

代回

sin (x) = \frac{3}{4}, sin (x) = - 2

sin (x) = \frac{3}{4}, sin (x) = - 2

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

sin (x) = \frac{3}{4}

使用反三角函数性质

sin (x) = \frac{3}{4}

sin (x) = \frac{3}{4}

的通解

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

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

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

sin (x) = - 2 :

无解

sin (x) = - 2

- 1 \leq sin (x) \leq 1

无解

合并所有解

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

以小数形式表示解

x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn

作图

流行的例子

csc(x)+cot(x)=0

csc (x) + cot (x) = 0

cos^2(θ)-sin^2(θ)=0

cos^{2} (θ) - sin^{2} (θ) = 0

7=4sin(pi/2 (x-2))+5

7 = 4 sin (\frac{π}{2} (x - 2)) + 5

5 - cos (θ) = 5

6sin(θ)=2+2sin(θ)

6 sin (θ) = 2 + 2 sin (θ)