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受欢迎的三角函数 >

13sin^2(θ)-7sin(θ)=5

解答

13 sin^{2} (θ) - 7 sin (θ) = 5

解答

θ = 1.23858 \dots + 2 πn, θ = π - 1.23858 \dots + 2 πn, θ = - 0.41901 \dots + 2 πn, θ = π + 0.41901 \dots + 2 πn

+1

度数

θ = 70.96559 \dots^{\circ} + 36 0^{\circ} n, θ = 109.03440 \dots^{\circ} + 36 0^{\circ} n, θ = - 24.00782 \dots^{\circ} + 36 0^{\circ} n, θ = 204.00782 \dots^{\circ} + 36 0^{\circ} n

求解步骤

13 sin^{2} (θ) - 7 sin (θ) = 5

用替代法求解

13 sin^{2} (θ) - 7 sin (θ) = 5

令

: sin (θ) = u

13 u^{2} - 7 u = 5

13 u^{2} - 7 u = 5 : u = \frac{7 + 309}{26}, u = \frac{7 - 309}{26}

13 u^{2} - 7 u = 5

将

5

para o lado esquerdo

13 u^{2} - 7 u = 5

两边减去

5

13 u^{2} - 7 u - 5 = 5 - 5

化简

13 u^{2} - 7 u - 5 = 0

13 u^{2} - 7 u - 5 = 0

使用求根公式求解

13 u^{2} - 7 u - 5 = 0

二次方程求根公式:

若

a = 13, b = - 7, c = - 5

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 13 ( - 5 )}{2 \cdot 13}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 13 ( - 5 )}{2 \cdot 13}

(- 7)^{2} - 4 \cdot 13 (- 5) = 309

(- 7)^{2} - 4 \cdot 13 (- 5)

使用法则

- (- a) = a

= (- 7)^{2} + 4 \cdot 13 \cdot 5

使用指数法则:

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

若

n

是偶数

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

= 7^{2} + 4 \cdot 13 \cdot 5

数字相乘:

4 \cdot 13 \cdot 5 = 260

= 7^{2} + 260

7^{2} = 49

= 49 + 260

数字相加:

49 + 260 = 309

= 309

u_{1, 2} = \frac{- ( - 7 ) \pm 309}{2 \cdot 13}

将解分隔开

u_{1} = \frac{- ( - 7 ) + 309}{2 \cdot 13}, u_{2} = \frac{- ( - 7 ) - 309}{2 \cdot 13}

u = \frac{- ( - 7 ) + 309}{2 \cdot 13} : \frac{7 + 309}{26}

\frac{- ( - 7 ) + 309}{2 \cdot 13}

使用法则

- (- a) = a

= \frac{7 + 309}{2 \cdot 13}

数字相乘:

2 \cdot 13 = 26

= \frac{7 + 309}{26}

u = \frac{- ( - 7 ) - 309}{2 \cdot 13} : \frac{7 - 309}{26}

\frac{- ( - 7 ) - 309}{2 \cdot 13}

使用法则

- (- a) = a

= \frac{7 - 309}{2 \cdot 13}

数字相乘:

2 \cdot 13 = 26

= \frac{7 - 309}{26}

二次方程组的解是:

u = \frac{7 + 309}{26}, u = \frac{7 - 309}{26}

u = sin (θ)

代回

sin (θ) = \frac{7 + 309}{26}, sin (θ) = \frac{7 - 309}{26}

sin (θ) = \frac{7 + 309}{26}, sin (θ) = \frac{7 - 309}{26}

sin (θ) = \frac{7 + 309}{26} : θ = arcsin (\frac{7 + 309}{26}) + 2 πn, θ = π - arcsin (\frac{7 + 309}{26}) + 2 πn

sin (θ) = \frac{7 + 309}{26}

使用反三角函数性质

sin (θ) = \frac{7 + 309}{26}

sin (θ) = \frac{7 + 309}{26}

的通解

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

θ = arcsin (\frac{7 + 309}{26}) + 2 πn, θ = π - arcsin (\frac{7 + 309}{26}) + 2 πn

θ = arcsin (\frac{7 + 309}{26}) + 2 πn, θ = π - arcsin (\frac{7 + 309}{26}) + 2 πn

sin (θ) = \frac{7 - 309}{26} : θ = arcsin (\frac{7 - 309}{26}) + 2 πn, θ = π + arcsin (- \frac{7 - 309}{26}) + 2 πn

sin (θ) = \frac{7 - 309}{26}

使用反三角函数性质

sin (θ) = \frac{7 - 309}{26}

sin (θ) = \frac{7 - 309}{26}

的通解

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

θ = arcsin (\frac{7 - 309}{26}) + 2 πn, θ = π + arcsin (- \frac{7 - 309}{26}) + 2 πn

θ = arcsin (\frac{7 - 309}{26}) + 2 πn, θ = π + arcsin (- \frac{7 - 309}{26}) + 2 πn

合并所有解

θ = arcsin (\frac{7 + 309}{26}) + 2 πn, θ = π - arcsin (\frac{7 + 309}{26}) + 2 πn, θ = arcsin (\frac{7 - 309}{26}) + 2 πn, θ = π + arcsin (- \frac{7 - 309}{26}) + 2 πn

以小数形式表示解

θ = 1.23858 \dots + 2 πn, θ = π - 1.23858 \dots + 2 πn, θ = - 0.41901 \dots + 2 πn, θ = π + 0.41901 \dots + 2 πn

作图

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