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受欢迎的三角函数 >

4cos(2x)+2sin(x)=1,0<= x<= 360

解答

4 cos (2 x) + 2 sin (x) = 1, 0^{\circ} \leq x \leq 36 0^{\circ}

解答

x = 21 0^{\circ}, x = 33 0^{\circ}, x = 0.84806 \dots, x = 18 0^{\circ} - 0.84806 \dots

+1

弧度

x = \frac{7 π}{6}, x = \frac{11 π}{6}, x = 0.84806 \dots, x = π - 0.84806 \dots

求解步骤

4 cos (2 x) + 2 sin (x) = 1, 0^{\circ} \leq x \leq 36 0^{\circ}

两边减去

1

4 cos (2 x) + 2 sin (x) - 1 = 0

使用三角恒等式改写

- 1 + 2 sin (x) + 4 cos (2 x)

使用倍角公式:

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

= - 1 + 2 sin (x) + 4 (1 - 2 sin^{2} (x))

化简

- 1 + 2 sin (x) + 4 (1 - 2 sin^{2} (x)) : 2 sin (x) - 8 sin^{2} (x) + 3

- 1 + 2 sin (x) + 4 (1 - 2 sin^{2} (x))

乘开

4 (1 - 2 sin^{2} (x)) : 4 - 8 sin^{2} (x)

4 (1 - 2 sin^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 4, b = 1, c = 2 sin^{2} (x)

= 4 \cdot 1 - 4 \cdot 2 sin^{2} (x)

化简

4 \cdot 1 - 4 \cdot 2 sin^{2} (x) : 4 - 8 sin^{2} (x)

4 \cdot 1 - 4 \cdot 2 sin^{2} (x)

数字相乘:

4 \cdot 1 = 4

= 4 - 4 \cdot 2 sin^{2} (x)

数字相乘:

4 \cdot 2 = 8

= 4 - 8 sin^{2} (x)

= 4 - 8 sin^{2} (x)

= - 1 + 2 sin (x) + 4 - 8 sin^{2} (x)

化简

- 1 + 2 sin (x) + 4 - 8 sin^{2} (x) : 2 sin (x) - 8 sin^{2} (x) + 3

- 1 + 2 sin (x) + 4 - 8 sin^{2} (x)

对同类项分组

= 2 sin (x) - 8 sin^{2} (x) - 1 + 4

数字相加/相减:

- 1 + 4 = 3

= 2 sin (x) - 8 sin^{2} (x) + 3

= 2 sin (x) - 8 sin^{2} (x) + 3

= 2 sin (x) - 8 sin^{2} (x) + 3

3 + 2 sin (x) - 8 sin^{2} (x) = 0

用替代法求解

3 + 2 sin (x) - 8 sin^{2} (x) = 0

令

: sin (x) = u

3 + 2 u - 8 u^{2} = 0

3 + 2 u - 8 u^{2} = 0 : u = - \frac{1}{2}, u = \frac{3}{4}

3 + 2 u - 8 u^{2} = 0

改写成标准形式

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

- 8 u^{2} + 2 u + 3 = 0

使用求根公式求解

- 8 u^{2} + 2 u + 3 = 0

二次方程求根公式:

若

a = - 8, b = 2, c = 3

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 ( - 8 ) \cdot 3}{2 ( - 8 )}

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 ( - 8 ) \cdot 3}{2 ( - 8 )}

2^{2} - 4 (- 8) \cdot 3 = 10

2^{2} - 4 (- 8) \cdot 3

使用法则

- (- a) = a

= 2^{2} + 4 \cdot 8 \cdot 3

数字相乘:

4 \cdot 8 \cdot 3 = 96

= 2^{2} + 96

2^{2} = 4

= 4 + 96

数字相加:

4 + 96 = 100

= 100

因式分解数字:

100 = 1 0^{2}

= 1 0^{2}

使用根式运算法则:

n a^{n} = a

1 0^{2} = 10

= 10

u_{1, 2} = \frac{- 2 \pm 10}{2 ( - 8 )}

将解分隔开

u_{1} = \frac{- 2 + 10}{2 ( - 8 )}, u_{2} = \frac{- 2 - 10}{2 ( - 8 )}

u = \frac{- 2 + 10}{2 ( - 8 )} : - \frac{1}{2}

\frac{- 2 + 10}{2 ( - 8 )}

去除括号:

(- a) = - a

= \frac{- 2 + 10}{- 2 \cdot 8}

数字相加/相减:

- 2 + 10 = 8

= \frac{8}{- 2 \cdot 8}

数字相乘:

2 \cdot 8 = 16

= \frac{8}{- 16}

使用分式法则:

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

= - \frac{8}{16}

约分:

8

= - \frac{1}{2}

u = \frac{- 2 - 10}{2 ( - 8 )} : \frac{3}{4}

\frac{- 2 - 10}{2 ( - 8 )}

去除括号:

(- a) = - a

= \frac{- 2 - 10}{- 2 \cdot 8}

数字相减:

- 2 - 10 = - 12

= \frac{- 12}{- 2 \cdot 8}

数字相乘:

2 \cdot 8 = 16

= \frac{- 12}{- 16}

使用分式法则:

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

= \frac{12}{16}

约分:

4

= \frac{3}{4}

二次方程组的解是:

u = - \frac{1}{2}, u = \frac{3}{4}

u = sin (x)

代回

sin (x) = - \frac{1}{2}, sin (x) = \frac{3}{4}

sin (x) = - \frac{1}{2}, sin (x) = \frac{3}{4}

sin (x) = - \frac{1}{2}, 0 \leq x \leq 36 0^{\circ} : x = 21 0^{\circ}, x = 33 0^{\circ}

sin (x) = - \frac{1}{2}, 0 \leq x \leq 36 0^{\circ}

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

的通解

sin (x)

周期表（周期为

36 0^{\circ} n

"）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

在

0 \leq x \leq 36 0^{\circ}

范围内的解

x = 21 0^{\circ}, x = 33 0^{\circ}

sin (x) = \frac{3}{4}, 0 \leq x \leq 36 0^{\circ} : x = arcsin (\frac{3}{4}), x = 18 0^{\circ} - arcsin (\frac{3}{4})

sin (x) = \frac{3}{4}, 0 \leq x \leq 36 0^{\circ}

使用反三角函数性质

sin (x) = \frac{3}{4}

sin (x) = \frac{3}{4}

的通解

sin (x) = a \Rightarrow x = arcsin (a) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (a) + 36 0^{\circ} n

x = arcsin (\frac{3}{4}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{3}{4}) + 36 0^{\circ} n

x = arcsin (\frac{3}{4}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{3}{4}) + 36 0^{\circ} n

在

0 \leq x \leq 36 0^{\circ}

范围内的解

x = arcsin (\frac{3}{4}), x = 18 0^{\circ} - arcsin (\frac{3}{4})

合并所有解

x = 21 0^{\circ}, x = 33 0^{\circ}, x = arcsin (\frac{3}{4}), x = 18 0^{\circ} - arcsin (\frac{3}{4})

以小数形式表示解

x = 21 0^{\circ}, x = 33 0^{\circ}, x = 0.84806 \dots, x = 18 0^{\circ} - 0.84806 \dots

作图

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