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受欢迎的三角函数 >

20sin^2(x)-21sin(x)+4=0

解答

20 sin^{2} (x) - 21 sin (x) + 4 = 0

解答

x = 0.92729 \dots + 2 πn, x = π - 0.92729 \dots + 2 πn, x = 0.25268 \dots + 2 πn, x = π - 0.25268 \dots + 2 πn

+1

度数

x = 53.13010 \dots^{\circ} + 36 0^{\circ} n, x = 126.86989 \dots^{\circ} + 36 0^{\circ} n, x = 14.47751 \dots^{\circ} + 36 0^{\circ} n, x = 165.52248 \dots^{\circ} + 36 0^{\circ} n

求解步骤

20 sin^{2} (x) - 21 sin (x) + 4 = 0

用替代法求解

20 sin^{2} (x) - 21 sin (x) + 4 = 0

令

: sin (x) = u

20 u^{2} - 21 u + 4 = 0

20 u^{2} - 21 u + 4 = 0 : u = \frac{4}{5}, u = \frac{1}{4}

20 u^{2} - 21 u + 4 = 0

使用求根公式求解

20 u^{2} - 21 u + 4 = 0

二次方程求根公式:

若

a = 20, b = - 21, c = 4

u_{1, 2} = \frac{- ( - 21 ) \pm ( - 21 ) ^{2} - 4 \cdot 20 \cdot 4}{2 \cdot 20}

u_{1, 2} = \frac{- ( - 21 ) \pm ( - 21 ) ^{2} - 4 \cdot 20 \cdot 4}{2 \cdot 20}

(- 21)^{2} - 4 \cdot 20 \cdot 4 = 11

(- 21)^{2} - 4 \cdot 20 \cdot 4

使用指数法则:

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

若

n

是偶数

(- 21)^{2} = 2 1^{2}

= 2 1^{2} - 4 \cdot 20 \cdot 4

数字相乘:

4 \cdot 20 \cdot 4 = 320

= 2 1^{2} - 320

2 1^{2} = 441

= 441 - 320

数字相减:

441 - 320 = 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{- ( - 21 ) \pm 11}{2 \cdot 20}

将解分隔开

u_{1} = \frac{- ( - 21 ) + 11}{2 \cdot 20}, u_{2} = \frac{- ( - 21 ) - 11}{2 \cdot 20}

u = \frac{- ( - 21 ) + 11}{2 \cdot 20} : \frac{4}{5}

\frac{- ( - 21 ) + 11}{2 \cdot 20}

使用法则

- (- a) = a

= \frac{21 + 11}{2 \cdot 20}

数字相加:

21 + 11 = 32

= \frac{32}{2 \cdot 20}

数字相乘:

2 \cdot 20 = 40

= \frac{32}{40}

约分:

8

= \frac{4}{5}

u = \frac{- ( - 21 ) - 11}{2 \cdot 20} : \frac{1}{4}

\frac{- ( - 21 ) - 11}{2 \cdot 20}

使用法则

- (- a) = a

= \frac{21 - 11}{2 \cdot 20}

数字相减:

21 - 11 = 10

= \frac{10}{2 \cdot 20}

数字相乘:

2 \cdot 20 = 40

= \frac{10}{40}

约分:

10

= \frac{1}{4}

二次方程组的解是:

u = \frac{4}{5}, u = \frac{1}{4}

u = sin (x)

代回

sin (x) = \frac{4}{5}, sin (x) = \frac{1}{4}

sin (x) = \frac{4}{5}, sin (x) = \frac{1}{4}

sin (x) = \frac{4}{5} : x = arcsin (\frac{4}{5}) + 2 πn, x = π - arcsin (\frac{4}{5}) + 2 πn

sin (x) = \frac{4}{5}

使用反三角函数性质

sin (x) = \frac{4}{5}

sin (x) = \frac{4}{5}

的通解

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

x = arcsin (\frac{4}{5}) + 2 πn, x = π - arcsin (\frac{4}{5}) + 2 πn

x = arcsin (\frac{4}{5}) + 2 πn, x = π - arcsin (\frac{4}{5}) + 2 πn

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{4}{5}) + 2 πn, x = π - arcsin (\frac{4}{5}) + 2 πn, x = arcsin (\frac{1}{4}) + 2 πn, x = π - arcsin (\frac{1}{4}) + 2 πn

以小数形式表示解

x = 0.92729 \dots + 2 πn, x = π - 0.92729 \dots + 2 πn, x = 0.25268 \dots + 2 πn, x = π - 0.25268 \dots + 2 πn

作图

流行的例子

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cot^{2} (θ) = - cot (θ)

0.2727=sec(2.746x)-1

0.2727 = sec (2.746 x) - 1

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