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受欢迎的三角函数 >

(1+tan(x))(5sin(x)-2)=0

解答

(1 + tan (x)) (5 sin (x) - 2) = 0

解答

x = \frac{3 π}{4} + πn, x = 0.41151 \dots + 2 πn, x = π - 0.41151 \dots + 2 πn

+1

度数

x = 13 5^{\circ} + 18 0^{\circ} n, x = 23.57817 \dots^{\circ} + 36 0^{\circ} n, x = 156.42182 \dots^{\circ} + 36 0^{\circ} n

求解步骤

(1 + tan (x)) (5 sin (x) - 2) = 0

分别求解每个部分

1 + tan (x) = 0 or 5 sin (x) - 2 = 0

1 + tan (x) = 0 : x = \frac{3 π}{4} + πn

1 + tan (x) = 0

将

1

到右边

1 + tan (x) = 0

两边减去

1

1 + tan (x) - 1 = 0 - 1

化简

tan (x) = - 1

tan (x) = - 1

tan (x) = - 1

的通解

tan (x)

周期表（周期为

πn

）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

5 sin (x) - 2 = 0 : x = arcsin (\frac{2}{5}) + 2 πn, x = π - arcsin (\frac{2}{5}) + 2 πn

5 sin (x) - 2 = 0

将

2

到右边

5 sin (x) - 2 = 0

两边加上

2

5 sin (x) - 2 + 2 = 0 + 2

化简

5 sin (x) = 2

5 sin (x) = 2

两边除以

5

5 sin (x) = 2

两边除以

5

\frac{5 sin ( x )}{5} = \frac{2}{5}

化简

sin (x) = \frac{2}{5}

sin (x) = \frac{2}{5}

使用反三角函数性质

sin (x) = \frac{2}{5}

sin (x) = \frac{2}{5}

的通解

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

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

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

合并所有解

x = \frac{3 π}{4} + πn, x = arcsin (\frac{2}{5}) + 2 πn, x = π - arcsin (\frac{2}{5}) + 2 πn

以小数形式表示解

x = \frac{3 π}{4} + πn, x = 0.41151 \dots + 2 πn, x = π - 0.41151 \dots + 2 πn

作图

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