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受欢迎的三角函数 >

21sin(t)cos(t)=-6cos(t)

解答

21 sin (t) cos (t) = - 6 cos (t)

解答

t = \frac{π}{2} + 2 πn, t = \frac{3 π}{2} + 2 πn, t = - 0.28975 \dots + 2 πn, t = π + 0.28975 \dots + 2 πn

+1

度数

t = 9 0^{\circ} + 36 0^{\circ} n, t = 27 0^{\circ} + 36 0^{\circ} n, t = - 16.60154 \dots^{\circ} + 36 0^{\circ} n, t = 196.60154 \dots^{\circ} + 36 0^{\circ} n

求解步骤

21 sin (t) cos (t) = - 6 cos (t)

两边减去

- 6 cos (t)

21 sin (t) cos (t) + 6 cos (t) = 0

分解

21 sin (t) cos (t) + 6 cos (t) : 3 cos (t) (7 sin (t) + 2)

21 sin (t) cos (t) + 6 cos (t)

将

6

改写为

2 \cdot 3

将

21

改写为

7 \cdot 3

= 7 \cdot 3 sin (t) cos (t) + 2 \cdot 3 cos (t)

因式分解出通项

3 cos (t)

= 3 cos (t) (7 sin (t) + 2)

3 cos (t) (7 sin (t) + 2) = 0

分别求解每个部分

cos (t) = 0 or 7 sin (t) + 2 = 0

cos (t) = 0 : t = \frac{π}{2} + 2 πn, t = \frac{3 π}{2} + 2 πn

cos (t) = 0

cos (t) = 0

的通解

cos (x)

周期表（周期为

2 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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

t = \frac{π}{2} + 2 πn, t = \frac{3 π}{2} + 2 πn

t = \frac{π}{2} + 2 πn, t = \frac{3 π}{2} + 2 πn

7 sin (t) + 2 = 0 : t = arcsin (- \frac{2}{7}) + 2 πn, t = π + arcsin (\frac{2}{7}) + 2 πn

7 sin (t) + 2 = 0

将

2

到右边

7 sin (t) + 2 = 0

两边减去

2

7 sin (t) + 2 - 2 = 0 - 2

化简

7 sin (t) = - 2

7 sin (t) = - 2

两边除以

7

7 sin (t) = - 2

两边除以

7

\frac{7 sin ( t )}{7} = \frac{- 2}{7}

化简

sin (t) = - \frac{2}{7}

sin (t) = - \frac{2}{7}

使用反三角函数性质

sin (t) = - \frac{2}{7}

sin (t) = - \frac{2}{7}

的通解

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

t = arcsin (- \frac{2}{7}) + 2 πn, t = π + arcsin (\frac{2}{7}) + 2 πn

t = arcsin (- \frac{2}{7}) + 2 πn, t = π + arcsin (\frac{2}{7}) + 2 πn

合并所有解

t = \frac{π}{2} + 2 πn, t = \frac{3 π}{2} + 2 πn, t = arcsin (- \frac{2}{7}) + 2 πn, t = π + arcsin (\frac{2}{7}) + 2 πn

以小数形式表示解

t = \frac{π}{2} + 2 πn, t = \frac{3 π}{2} + 2 πn, t = - 0.28975 \dots + 2 πn, t = π + 0.28975 \dots + 2 πn

作图

流行的例子

sec (θ) = 4

5 sin (x) = 1

5 sin (x) = 5

(1+sin(x))(1+sin(x))=cos^2(x)

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

cos (θ) = - 0.866