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受欢迎的三角函数 >

sin(5x)+sin(7x)=0,-90<= x<90

解答

sin (5 x) + sin (7 x) = 0, - 9 0^{\circ} \leq x < 9 0^{\circ}

解答

x = - 9 0^{\circ}, x = 0, x = 3 0^{\circ}, x = 6 0^{\circ}, x = - 6 0^{\circ}, x = - 3 0^{\circ}

+1

弧度

x = - \frac{π}{2}, x = 0, x = \frac{π}{6}, x = \frac{π}{3}, x = - \frac{π}{3}, x = - \frac{π}{6}

求解步骤

sin (5 x) + sin (7 x) = 0, - 9 0^{\circ} \leq x < 9 0^{\circ}

使用三角恒等式改写

sin (5 x) + sin (7 x)

使用和差化积恒等式:

sin (s) + sin (t) = 2 sin (\frac{s + t}{2}) cos (\frac{s - t}{2})

= 2 sin (\frac{5 x + 7 x}{2}) cos (\frac{5 x - 7 x}{2})

化简

2 sin (\frac{5 x + 7 x}{2}) cos (\frac{5 x - 7 x}{2}) : 2 cos (x) sin (6 x)

2 sin (\frac{5 x + 7 x}{2}) cos (\frac{5 x - 7 x}{2})

\frac{5 x + 7 x}{2} = 6 x

\frac{5 x + 7 x}{2}

同类项相加:

5 x + 7 x = 12 x

= \frac{12 x}{2}

数字相除:

\frac{12}{2} = 6

= 6 x

= 2 sin (6 x) cos (\frac{5 x - 7 x}{2})

\frac{5 x - 7 x}{2} = - x

\frac{5 x - 7 x}{2}

同类项相加:

5 x - 7 x = - 2 x

= \frac{- 2 x}{2}

使用分式法则:

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

= - \frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= - x

= 2 sin (6 x) cos (- x)

使用负角恒等式:

cos (- x) = cos (x)

= 2 cos (x) sin (6 x)

= 2 cos (x) sin (6 x)

2 cos (x) sin (6 x) = 0

分别求解每个部分

cos (x) = 0 or sin (6 x) = 0

cos (x) = 0, - 9 0^{\circ} \leq x < 9 0^{\circ} : x = - 9 0^{\circ}

cos (x) = 0, - 9 0^{\circ} \leq x < 9 0^{\circ}

cos (x) = 0

的通解

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

在

- 9 0^{\circ} \leq x < 9 0^{\circ}

范围内的解

x = - 9 0^{\circ}

sin (6 x) = 0, - 9 0^{\circ} \leq x < 9 0^{\circ} : x = 0, x = 3 0^{\circ}, x = 6 0^{\circ}, x = - 6 0^{\circ}, x = - 3 0^{\circ}, x = - 9 0^{\circ}

sin (6 x) = 0, - 9 0^{\circ} \leq x < 9 0^{\circ}

sin (6 x) = 0

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}

6 x = 0 + 36 0^{\circ} n, 6 x = 18 0^{\circ} + 36 0^{\circ} n

6 x = 0 + 36 0^{\circ} n, 6 x = 18 0^{\circ} + 36 0^{\circ} n

解

6 x = 0 + 36 0^{\circ} n : x = \frac{18 0 ^{\circ} n}{3}

6 x = 0 + 36 0^{\circ} n

0 + 36 0^{\circ} n = 36 0^{\circ} n

6 x = 36 0^{\circ} n

两边除以

6

6 x = 36 0^{\circ} n

两边除以

6

\frac{6 x}{6} = \frac{36 0 ^{\circ} n}{6}

化简

x = \frac{18 0 ^{\circ} n}{3}

x = \frac{18 0 ^{\circ} n}{3}

解

6 x = 18 0^{\circ} + 36 0^{\circ} n : x = 3 0^{\circ} + \frac{18 0 ^{\circ} n}{3}

6 x = 18 0^{\circ} + 36 0^{\circ} n

两边除以

6

6 x = 18 0^{\circ} + 36 0^{\circ} n

两边除以

6

\frac{6 x}{6} = 3 0^{\circ} + \frac{36 0 ^{\circ} n}{6}

化简

x = 3 0^{\circ} + \frac{18 0 ^{\circ} n}{3}

x = 3 0^{\circ} + \frac{18 0 ^{\circ} n}{3}

x = \frac{18 0 ^{\circ} n}{3}, x = 3 0^{\circ} + \frac{18 0 ^{\circ} n}{3}

在

- 9 0^{\circ} \leq x < 9 0^{\circ}

范围内的解

x = 0, x = 3 0^{\circ}, x = 6 0^{\circ}, x = - 6 0^{\circ}, x = - 3 0^{\circ}, x = - 9 0^{\circ}

合并所有解

x = - 9 0^{\circ}, x = 0, x = 3 0^{\circ}, x = 6 0^{\circ}, x = - 6 0^{\circ}, x = - 3 0^{\circ}

作图

流行的例子

-cos(θ)+1=2sin^2(θ)

- cos (θ) + 1 = 2 sin^{2} (θ)

arcsin(x)=(3pi)/7

arcsin (x) = \frac{3 π}{7}

tan (a) = \frac{3}{4}

3 tan (x) + 3 = 0

cos (π x) = 0.5