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受欢迎的三角函数 >

9sin(x)-2=4sin^2(x)

解答

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

解答

x = 0.25268 \dots + 2 πn, x = π - 0.25268 \dots + 2 πn

+1

度数

x = 14.47751 \dots^{\circ} + 36 0^{\circ} n, x = 165.52248 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

用替代法求解

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

令

: sin (x) = u

9 u - 2 = 4 u^{2}

9 u - 2 = 4 u^{2} : u = 2, u = \frac{1}{4}

9 u - 2 = 4 u^{2}

交换两边

4 u^{2} = 9 u - 2

将

2

para o lado esquerdo

4 u^{2} = 9 u - 2

两边加上

2

4 u^{2} + 2 = 9 u - 2 + 2

化简

4 u^{2} + 2 = 9 u

4 u^{2} + 2 = 9 u

将

9 u

para o lado esquerdo

4 u^{2} + 2 = 9 u

两边减去

9 u

4 u^{2} + 2 - 9 u = 9 u - 9 u

化简

4 u^{2} + 2 - 9 u = 0

4 u^{2} + 2 - 9 u = 0

改写成标准形式

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

4 u^{2} - 9 u + 2 = 0

使用求根公式求解

4 u^{2} - 9 u + 2 = 0

二次方程求根公式:

若

a = 4, b = - 9, c = 2

u_{1, 2} = \frac{- ( - 9 ) \pm ( - 9 ) ^{2} - 4 \cdot 4 \cdot 2}{2 \cdot 4}

u_{1, 2} = \frac{- ( - 9 ) \pm ( - 9 ) ^{2} - 4 \cdot 4 \cdot 2}{2 \cdot 4}

(- 9)^{2} - 4 \cdot 4 \cdot 2 = 7

(- 9)^{2} - 4 \cdot 4 \cdot 2

使用指数法则:

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

若

n

是偶数

(- 9)^{2} = 9^{2}

= 9^{2} - 4 \cdot 4 \cdot 2

数字相乘:

4 \cdot 4 \cdot 2 = 32

= 9^{2} - 32

9^{2} = 81

= 81 - 32

数字相减:

81 - 32 = 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{- ( - 9 ) \pm 7}{2 \cdot 4}

将解分隔开

u_{1} = \frac{- ( - 9 ) + 7}{2 \cdot 4}, u_{2} = \frac{- ( - 9 ) - 7}{2 \cdot 4}

u = \frac{- ( - 9 ) + 7}{2 \cdot 4} : 2

\frac{- ( - 9 ) + 7}{2 \cdot 4}

使用法则

- (- a) = a

= \frac{9 + 7}{2 \cdot 4}

数字相加:

9 + 7 = 16

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

数字相乘:

2 \cdot 4 = 8

= \frac{16}{8}

数字相除:

\frac{16}{8} = 2

= 2

u = \frac{- ( - 9 ) - 7}{2 \cdot 4} : \frac{1}{4}

\frac{- ( - 9 ) - 7}{2 \cdot 4}

使用法则

- (- a) = a

= \frac{9 - 7}{2 \cdot 4}

数字相减:

9 - 7 = 2

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

数字相乘:

2 \cdot 4 = 8

= \frac{2}{8}

约分:

2

= \frac{1}{4}

二次方程组的解是:

u = 2, u = \frac{1}{4}

u = sin (x)

代回

sin (x) = 2, sin (x) = \frac{1}{4}

sin (x) = 2, sin (x) = \frac{1}{4}

sin (x) = 2 :

无解

sin (x) = 2

- 1 \leq sin (x) \leq 1

无解

sin (x) = \frac{1}{4} : x = arcsin (\frac{1}{4}) + 2 πn, x = π - arcsin (\frac{1}{4}) + 2 πn

sin (x) = \frac{1}{4}

使用反三角函数性质

sin (x) = \frac{1}{4}

sin (x) = \frac{1}{4}

的通解

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

x = arcsin (\frac{1}{4}) + 2 πn, x = π - arcsin (\frac{1}{4}) + 2 πn

x = arcsin (\frac{1}{4}) + 2 πn, x = π - arcsin (\frac{1}{4}) + 2 πn

合并所有解

x = arcsin (\frac{1}{4}) + 2 πn, x = π - arcsin (\frac{1}{4}) + 2 πn

以小数形式表示解

x = 0.25268 \dots + 2 πn, x = π - 0.25268 \dots + 2 πn

作图

流行的例子

-2cot(x)=csc^2(x)

- 2 cot (x) = csc^{2} (x)

sin (θ) = - 0.671

(sin^2(x)+5cos^2(x))/2 =2

\frac{sin ^{2} ( x ) + 5 cos ^{2} ( x )}{2} = 2

2sin(x)=7-3csc(x)

2 sin (x) = 7 - 3 csc (x)

cos(θ)=(134/109)

cos (θ) = (\frac{134}{109})