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受欢迎的三角函数 >

3cos^2(x)=7-8sin(x)

解答

3 cos^{2} (x) = 7 - 8 sin (x)

解答

x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

+1

度数

x = 41.81031 \dots^{\circ} + 36 0^{\circ} n, x = 138.18968 \dots^{\circ} + 36 0^{\circ} n

求解步骤

3 cos^{2} (x) = 7 - 8 sin (x)

两边减去

7 - 8 sin (x)

3 cos^{2} (x) - 7 + 8 sin (x) = 0

使用三角恒等式改写

- 7 + 3 cos^{2} (x) + 8 sin (x)

使用毕达哥拉斯恒等式:

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

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

= - 7 + 3 (1 - sin^{2} (x)) + 8 sin (x)

化简

- 7 + 3 (1 - sin^{2} (x)) + 8 sin (x) : 8 sin (x) - 3 sin^{2} (x) - 4

- 7 + 3 (1 - sin^{2} (x)) + 8 sin (x)

乘开

3 (1 - sin^{2} (x)) : 3 - 3 sin^{2} (x)

3 (1 - sin^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 3, b = 1, c = sin^{2} (x)

= 3 \cdot 1 - 3 sin^{2} (x)

数字相乘:

3 \cdot 1 = 3

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

= - 7 + 3 - 3 sin^{2} (x) + 8 sin (x)

数字相加/相减:

- 7 + 3 = - 4

= 8 sin (x) - 3 sin^{2} (x) - 4

= 8 sin (x) - 3 sin^{2} (x) - 4

- 4 - 3 sin^{2} (x) + 8 sin (x) = 0

用替代法求解

- 4 - 3 sin^{2} (x) + 8 sin (x) = 0

令

: sin (x) = u

- 4 - 3 u^{2} + 8 u = 0

- 4 - 3 u^{2} + 8 u = 0 : u = \frac{2}{3}, u = 2

- 4 - 3 u^{2} + 8 u = 0

改写成标准形式

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

- 3 u^{2} + 8 u - 4 = 0

使用求根公式求解

- 3 u^{2} + 8 u - 4 = 0

二次方程求根公式:

若

a = - 3, b = 8, c = - 4

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 3 ) ( - 4 )}{2 ( - 3 )}

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 3 ) ( - 4 )}{2 ( - 3 )}

8^{2} - 4 (- 3) (- 4) = 4

8^{2} - 4 (- 3) (- 4)

使用法则

- (- a) = a

= 8^{2} - 4 \cdot 3 \cdot 4

数字相乘:

4 \cdot 3 \cdot 4 = 48

= 8^{2} - 48

8^{2} = 64

= 64 - 48

数字相减:

64 - 48 = 16

= 16

因式分解数字:

16 = 4^{2}

= 4^{2}

使用根式运算法则:

n a^{n} = a

4^{2} = 4

= 4

u_{1, 2} = \frac{- 8 \pm 4}{2 ( - 3 )}

将解分隔开

u_{1} = \frac{- 8 + 4}{2 ( - 3 )}, u_{2} = \frac{- 8 - 4}{2 ( - 3 )}

u = \frac{- 8 + 4}{2 ( - 3 )} : \frac{2}{3}

\frac{- 8 + 4}{2 ( - 3 )}

去除括号:

(- a) = - a

= \frac{- 8 + 4}{- 2 \cdot 3}

数字相加/相减:

- 8 + 4 = - 4

= \frac{- 4}{- 2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{- 4}{- 6}

使用分式法则:

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

= \frac{4}{6}

约分:

2

= \frac{2}{3}

u = \frac{- 8 - 4}{2 ( - 3 )} : 2

\frac{- 8 - 4}{2 ( - 3 )}

去除括号:

(- a) = - a

= \frac{- 8 - 4}{- 2 \cdot 3}

数字相减:

- 8 - 4 = - 12

= \frac{- 12}{- 2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{- 12}{- 6}

使用分式法则:

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

= \frac{12}{6}

数字相除:

\frac{12}{6} = 2

= 2

二次方程组的解是:

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

u = sin (x)

代回

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

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

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

sin (x) = \frac{2}{3}

使用反三角函数性质

sin (x) = \frac{2}{3}

sin (x) = \frac{2}{3}

的通解

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

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

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

以小数形式表示解

x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

作图

流行的例子

sin (X) = - \frac{1}{2}

-2(sin(2t)-cos(t))=0

- 2 (sin (2 t) - cos (t)) = 0

5cos(A)+8=3cos(A)+6

5 cos (A) + 8 = 3 cos (A) + 6

2+7cos(θ)=4sin^2(θ),-180<= x<= 180

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