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受欢迎的三角函数 >

(sin(x)+sin^2(x))/2 =0.5

解答

\frac{sin ( x ) + sin ^{2} ( x )}{2} = 0.5

解答

x = 0.66623 \dots + 2 πn, x = π - 0.66623 \dots + 2 πn

+1

度数

x = 38.17270 \dots^{\circ} + 36 0^{\circ} n, x = 141.82729 \dots^{\circ} + 36 0^{\circ} n

求解步骤

\frac{sin ( x ) + sin ^{2} ( x )}{2} = 0.5

用替代法求解

\frac{sin ( x ) + sin ^{2} ( x )}{2} = 0.5

令

: sin (x) = u

\frac{u + u ^{2}}{2} = 0.5

\frac{u + u ^{2}}{2} = 0.5 : u = \frac{- 1 + 5}{2}, u = - \frac{1 + 5}{2}

\frac{u + u ^{2}}{2} = 0.5

在两边乘以

10

\frac{u + u ^{2}}{2} = 0.5

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

\frac{u + u ^{2}}{2} \cdot 10 = 0.5 \cdot 10

整理后得

5 (u^{2} + u) = 5

5 (u^{2} + u) = 5

展开

5 (u^{2} + u) : 5 u^{2} + 5 u

5 (u^{2} + u)

使用分配律:

a (b + c) = ab + a c

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

= 5 u^{2} + 5 u

5 u^{2} + 5 u = 5

将

5

para o lado esquerdo

5 u^{2} + 5 u = 5

两边减去

5

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

化简

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = 5, b = 5, c = - 5

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

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

5^{2} - 4 \cdot 5 (- 5) = 55

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

使用法则

- (- a) = a

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

数字相乘:

4 \cdot 5 \cdot 5 = 100

= 5^{2} + 100

5^{2} = 25

= 25 + 100

数字相加:

25 + 100 = 125

= 125

125

质因数分解:

5^{3}

125

125

除以

5125 = 25 \cdot 5

= 5 \cdot 25

25

除以

525 = 5 \cdot 5

= 5 \cdot 5 \cdot 5

5

是质数，因此无法进一步因数分解

= 5 \cdot 5 \cdot 5

= 5^{3}

= 5^{3}

使用指数法则:

a^{b + c} = a^{b} \cdot a^{c}

= 5^{2} \cdot 5

使用根式运算法则:

n ab = n a n b

= 5 5^{2}

使用根式运算法则:

n a^{n} = a

5^{2} = 5

= 55

u_{1, 2} = \frac{- 5 \pm 5 5}{2 \cdot 5}

将解分隔开

u_{1} = \frac{- 5 + 5 5}{2 \cdot 5}, u_{2} = \frac{- 5 - 5 5}{2 \cdot 5}

u = \frac{- 5 + 5 5}{2 \cdot 5} : \frac{- 1 + 5}{2}

\frac{- 5 + 5 5}{2 \cdot 5}

数字相乘:

2 \cdot 5 = 10

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

分解

- 5 + 55 : 5 (- 1 + 5)

- 5 + 55

改写为

= - 5 \cdot 1 + 55

因式分解出通项

5

= 5 (- 1 + 5)

= \frac{5 ( - 1 + 5 )}{10}

约分:

5

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

u = \frac{- 5 - 5 5}{2 \cdot 5} : - \frac{1 + 5}{2}

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

数字相乘:

2 \cdot 5 = 10

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

分解

- 5 - 55 : - 5 (1 + 5)

- 5 - 55

改写为

= - 5 \cdot 1 - 55

因式分解出通项

5

= - 5 (1 + 5)

= - \frac{5 ( 1 + 5 )}{10}

约分:

5

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

二次方程组的解是:

u = \frac{- 1 + 5}{2}, u = - \frac{1 + 5}{2}

u = sin (x)

代回

sin (x) = \frac{- 1 + 5}{2}, sin (x) = - \frac{1 + 5}{2}

sin (x) = \frac{- 1 + 5}{2}, sin (x) = - \frac{1 + 5}{2}

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

sin (x) = \frac{- 1 + 5}{2}

使用反三角函数性质

sin (x) = \frac{- 1 + 5}{2}

sin (x) = \frac{- 1 + 5}{2}

的通解

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

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

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

sin (x) = - \frac{1 + 5}{2} :

无解

sin (x) = - \frac{1 + 5}{2}

- 1 \leq sin (x) \leq 1

无解

合并所有解

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

以小数形式表示解

x = 0.66623 \dots + 2 πn, x = π - 0.66623 \dots + 2 πn

作图

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