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受欢迎的三角函数 >

sin(θ)=-(sqrt(5))/3 cos(2θ)

解答

sin (θ) = - \frac{5}{3} cos (2 θ)

解答

θ = - 0.46364 \dots + 2 πn, θ = π + 0.46364 \dots + 2 πn

+1

度数

θ = - 26.56505 \dots^{\circ} + 36 0^{\circ} n, θ = 206.56505 \dots^{\circ} + 36 0^{\circ} n

求解步骤

sin (θ) = - \frac{5}{3} cos (2 θ)

两边减去

- \frac{5}{3} cos (2 θ)

sin (θ) + \frac{5}{3} cos (2 θ) = 0

化简

sin (θ) + \frac{5}{3} cos (2 θ) : \frac{3 sin ( θ ) + 5 cos ( 2 θ )}{3}

sin (θ) + \frac{5}{3} cos (2 θ)

乘

\frac{5}{3} cos (2 θ) : \frac{5 cos ( 2 θ )}{3}

\frac{5}{3} cos (2 θ)

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{5 cos ( 2 θ )}{3}

= sin (θ) + \frac{5 cos ( 2 θ )}{3}

将项转换为分式:

sin (θ) = \frac{sin ( θ ) 3}{3}

= \frac{sin ( θ ) \cdot 3}{3} + \frac{5 cos ( 2 θ )}{3}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{sin ( θ ) \cdot 3 + 5 cos ( 2 θ )}{3}

\frac{3 sin ( θ ) + 5 cos ( 2 θ )}{3} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

3 sin (θ) + 5 cos (2 θ) = 0

使用三角恒等式改写

3 sin (θ) + cos (2 θ) 5

使用倍角公式:

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

= 3 sin (θ) + 5 (1 - 2 sin^{2} (θ))

(1 - 2 sin^{2} (θ)) 5 + 3 sin (θ) = 0

用替代法求解

(1 - 2 sin^{2} (θ)) 5 + 3 sin (θ) = 0

令

: sin (θ) = u

(1 - 2 u^{2}) 5 + 3 u = 0

(1 - 2 u^{2}) 5 + 3 u = 0 : u = - \frac{5}{5}, u = \frac{5}{2}

(1 - 2 u^{2}) 5 + 3 u = 0

展开

(1 - 2 u^{2}) 5 + 3 u : 5 - 25 u^{2} + 3 u

(1 - 2 u^{2}) 5 + 3 u

= 5 (1 - 2 u^{2}) + 3 u

乘开

5 (1 - 2 u^{2}) : 5 - 25 u^{2}

5 (1 - 2 u^{2})

使用分配律:

a (b - c) = ab - a c

a = 5, b = 1, c = 2 u^{2}

= 5 \cdot 1 - 5 \cdot 2 u^{2}

= 1 \cdot 5 - 25 u^{2}

乘以:

1 \cdot 5 = 5

= 5 - 25 u^{2}

= 5 - 25 u^{2} + 3 u

5 - 25 u^{2} + 3 u = 0

改写成标准形式

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

- 25 u^{2} + 3 u + 5 = 0

使用求根公式求解

- 25 u^{2} + 3 u + 5 = 0

二次方程求根公式:

若

a = - 25, b = 3, c = 5

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 ( - 2 5 ) 5}{2 ( - 2 5 )}

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 ( - 2 5 ) 5}{2 ( - 2 5 )}

3^{2} - 4 (- 25) 5 = 7

3^{2} - 4 (- 25) 5

使用法则

- (- a) = a

= 3^{2} + 4 \cdot 255

4 \cdot 255 = 40

4 \cdot 255

数字相乘:

4 \cdot 2 = 8

= 855

使用根式运算法则:

a a = a

55 = 5

= 8 \cdot 5

数字相乘:

8 \cdot 5 = 40

= 40

= 3^{2} + 40

3^{2} = 9

= 9 + 40

数字相加:

9 + 40 = 49

= 49

因式分解数字:

49 = 7^{2}

= 7^{2}

使用根式运算法则:

n a^{n} = a

7^{2} = 7

= 7

u_{1, 2} = \frac{- 3 \pm 7}{2 ( - 2 5 )}

将解分隔开

u_{1} = \frac{- 3 + 7}{2 ( - 2 5 )}, u_{2} = \frac{- 3 - 7}{2 ( - 2 5 )}

u = \frac{- 3 + 7}{2 ( - 2 5 )} : - \frac{5}{5}

\frac{- 3 + 7}{2 ( - 2 5 )}

去除括号:

(- a) = - a

= \frac{- 3 + 7}{- 2 \cdot 2 5}

数字相加/相减:

- 3 + 7 = 4

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

数字相乘:

2 \cdot 2 = 4

= \frac{4}{- 4 5}

使用分式法则:

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

= - \frac{4}{4 5}

数字相除:

\frac{4}{4} = 1

= - \frac{1}{5}

- \frac{1}{5}

有理化:

- \frac{5}{5}

- \frac{1}{5}

乘以共轭根式

\frac{5}{5}

= - \frac{1 \cdot 5}{5 5}

1 \cdot 5 = 5

55 = 5

55

使用根式运算法则:

a a = a

55 = 5

= 5

= - \frac{5}{5}

= - \frac{5}{5}

u = \frac{- 3 - 7}{2 ( - 2 5 )} : \frac{5}{2}

\frac{- 3 - 7}{2 ( - 2 5 )}

去除括号:

(- a) = - a

= \frac{- 3 - 7}{- 2 \cdot 2 5}

数字相减:

- 3 - 7 = - 10

= \frac{- 10}{- 2 \cdot 2 5}

数字相乘:

2 \cdot 2 = 4

= \frac{- 10}{- 4 5}

使用分式法则:

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

= \frac{10}{4 5}

约分:

2

= \frac{5}{2 5}

使用根式运算法则:

n a = a^{\frac{1}{n}}

5 = 5^{\frac{1}{2}}

= \frac{5}{2 \cdot 5 ^{\frac{1}{2}}}

使用指数法则:

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

\frac{5 ^{1}}{5 ^{\frac{1}{2}}} = 5^{1 - \frac{1}{2}}

= \frac{5 ^{1 - \frac{1}{2}}}{2}

数字相减:

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

= \frac{5 ^{\frac{1}{2}}}{2}

使用根式运算法则:

a^{\frac{1}{n}} = n a

5^{\frac{1}{2}} = 5

= \frac{5}{2}

二次方程组的解是:

u = - \frac{5}{5}, u = \frac{5}{2}

u = sin (θ)

代回

sin (θ) = - \frac{5}{5}, sin (θ) = \frac{5}{2}

sin (θ) = - \frac{5}{5}, sin (θ) = \frac{5}{2}

sin (θ) = - \frac{5}{5} : θ = arcsin (- \frac{5}{5}) + 2 πn, θ = π + arcsin (\frac{5}{5}) + 2 πn

sin (θ) = - \frac{5}{5}

使用反三角函数性质

sin (θ) = - \frac{5}{5}

sin (θ) = - \frac{5}{5}

的通解

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

θ = arcsin (- \frac{5}{5}) + 2 πn, θ = π + arcsin (\frac{5}{5}) + 2 πn

θ = arcsin (- \frac{5}{5}) + 2 πn, θ = π + arcsin (\frac{5}{5}) + 2 πn

sin (θ) = \frac{5}{2} :

无解

sin (θ) = \frac{5}{2}

- 1 \leq sin (x) \leq 1

无解

合并所有解

θ = arcsin (- \frac{5}{5}) + 2 πn, θ = π + arcsin (\frac{5}{5}) + 2 πn

以小数形式表示解

θ = - 0.46364 \dots + 2 πn, θ = π + 0.46364 \dots + 2 πn

作图

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