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受欢迎的三角函数 >

-3cos(2θ)+5=7sin(θ)+5

解答

- 3 cos (2 θ) + 5 = 7 sin (θ) + 5

解答

θ = - 0.33983 \dots + 2 πn, θ = π + 0.33983 \dots + 2 πn

+1

度数

θ = - 19.47122 \dots^{\circ} + 36 0^{\circ} n, θ = 199.47122 \dots^{\circ} + 36 0^{\circ} n

求解步骤

- 3 cos (2 θ) + 5 = 7 sin (θ) + 5

两边减去

7 sin (θ) + 5

- 3 cos (2 θ) - 7 sin (θ) = 0

使用三角恒等式改写

- 3 cos (2 θ) - 7 sin (θ)

使用倍角公式:

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

= - 3 (1 - 2 sin^{2} (θ)) - 7 sin (θ)

- (1 - 2 sin^{2} (θ)) \cdot 3 - 7 sin (θ) = 0

用替代法求解

- (1 - 2 sin^{2} (θ)) \cdot 3 - 7 sin (θ) = 0

令

: sin (θ) = u

- (1 - 2 u^{2}) \cdot 3 - 7 u = 0

- (1 - 2 u^{2}) \cdot 3 - 7 u = 0 : u = \frac{3}{2}, u = - \frac{1}{3}

- (1 - 2 u^{2}) \cdot 3 - 7 u = 0

展开

- (1 - 2 u^{2}) \cdot 3 - 7 u : - 3 + 6 u^{2} - 7 u

- (1 - 2 u^{2}) \cdot 3 - 7 u

= - 3 (1 - 2 u^{2}) - 7 u

乘开

- 3 (1 - 2 u^{2}) : - 3 + 6 u^{2}

- 3 (1 - 2 u^{2})

使用分配律:

a (b - c) = ab - a c

a = - 3, b = 1, c = 2 u^{2}

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

使用加减运算法则

- (- a) = a

= - 3 \cdot 1 + 3 \cdot 2 u^{2}

化简

- 3 \cdot 1 + 3 \cdot 2 u^{2} : - 3 + 6 u^{2}

- 3 \cdot 1 + 3 \cdot 2 u^{2}

数字相乘:

3 \cdot 1 = 3

= - 3 + 3 \cdot 2 u^{2}

数字相乘:

3 \cdot 2 = 6

= - 3 + 6 u^{2}

= - 3 + 6 u^{2}

= - 3 + 6 u^{2} - 7 u

- 3 + 6 u^{2} - 7 u = 0

改写成标准形式

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

6 u^{2} - 7 u - 3 = 0

使用求根公式求解

6 u^{2} - 7 u - 3 = 0

二次方程求根公式:

若

a = 6, b = - 7, c = - 3

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 6 ( - 3 )}{2 \cdot 6}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 6 ( - 3 )}{2 \cdot 6}

(- 7)^{2} - 4 \cdot 6 (- 3) = 11

(- 7)^{2} - 4 \cdot 6 (- 3)

使用法则

- (- a) = a

= (- 7)^{2} + 4 \cdot 6 \cdot 3

使用指数法则:

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

若

n

是偶数

(- 7)^{2} = 7^{2}

= 7^{2} + 4 \cdot 6 \cdot 3

数字相乘:

4 \cdot 6 \cdot 3 = 72

= 7^{2} + 72

7^{2} = 49

= 49 + 72

数字相加:

49 + 72 = 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{- ( - 7 ) \pm 11}{2 \cdot 6}

将解分隔开

u_{1} = \frac{- ( - 7 ) + 11}{2 \cdot 6}, u_{2} = \frac{- ( - 7 ) - 11}{2 \cdot 6}

u = \frac{- ( - 7 ) + 11}{2 \cdot 6} : \frac{3}{2}

\frac{- ( - 7 ) + 11}{2 \cdot 6}

使用法则

- (- a) = a

= \frac{7 + 11}{2 \cdot 6}

数字相加:

7 + 11 = 18

= \frac{18}{2 \cdot 6}

数字相乘:

2 \cdot 6 = 12

= \frac{18}{12}

约分:

6

= \frac{3}{2}

u = \frac{- ( - 7 ) - 11}{2 \cdot 6} : - \frac{1}{3}

\frac{- ( - 7 ) - 11}{2 \cdot 6}

使用法则

- (- a) = a

= \frac{7 - 11}{2 \cdot 6}

数字相减:

7 - 11 = - 4

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

数字相乘:

2 \cdot 6 = 12

= \frac{- 4}{12}

使用分式法则:

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

= - \frac{4}{12}

约分:

4

= - \frac{1}{3}

二次方程组的解是:

u = \frac{3}{2}, u = - \frac{1}{3}

u = sin (θ)

代回

sin (θ) = \frac{3}{2}, sin (θ) = - \frac{1}{3}

sin (θ) = \frac{3}{2}, sin (θ) = - \frac{1}{3}

sin (θ) = \frac{3}{2} :

无解

sin (θ) = \frac{3}{2}

- 1 \leq sin (x) \leq 1

无解

sin (θ) = - \frac{1}{3} : θ = arcsin (- \frac{1}{3}) + 2 πn, θ = π + arcsin (\frac{1}{3}) + 2 πn

sin (θ) = - \frac{1}{3}

使用反三角函数性质

sin (θ) = - \frac{1}{3}

sin (θ) = - \frac{1}{3}

的通解

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

θ = arcsin (- \frac{1}{3}) + 2 πn, θ = π + arcsin (\frac{1}{3}) + 2 πn

θ = arcsin (- \frac{1}{3}) + 2 πn, θ = π + arcsin (\frac{1}{3}) + 2 πn

合并所有解

θ = arcsin (- \frac{1}{3}) + 2 πn, θ = π + arcsin (\frac{1}{3}) + 2 πn

以小数形式表示解

θ = - 0.33983 \dots + 2 πn, θ = π + 0.33983 \dots + 2 πn

作图

流行的例子

3+3cos(x)=2sin^2(x)

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

cos(θ)=-sin(θ)

cos (θ) = - sin (θ)

sqrt(2)csc(x)+2=4

2 csc (x) + 2 = 4

sec^2(θ)+5sec(θ)-6=0

sec^{2} (θ) + 5 sec (θ) - 6 = 0

cos (x) = 0.75