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受欢迎的三角函数 >

20sin(t)cos(t)=-16cos(t)

解答

20 sin (t) cos (t) = - 16 cos (t)

解答

t = \frac{π}{2} + 2 πn, t = \frac{3 π}{2} + 2 πn, t = - 0.92729 \dots + 2 πn, t = π + 0.92729 \dots + 2 πn

+1

度数

t = 9 0^{\circ} + 36 0^{\circ} n, t = 27 0^{\circ} + 36 0^{\circ} n, t = - 53.13010 \dots^{\circ} + 36 0^{\circ} n, t = 233.13010 \dots^{\circ} + 36 0^{\circ} n

求解步骤

20 sin (t) cos (t) = - 16 cos (t)

两边减去

- 16 cos (t)

20 sin (t) cos (t) + 16 cos (t) = 0

分解

20 sin (t) cos (t) + 16 cos (t) : 4 cos (t) (5 sin (t) + 4)

20 sin (t) cos (t) + 16 cos (t)

将

16

改写为

4 \cdot 4

将

20

改写为

5 \cdot 4

= 5 \cdot 4 sin (t) cos (t) + 4 \cdot 4 cos (t)

因式分解出通项

4 cos (t)

= 4 cos (t) (5 sin (t) + 4)

4 cos (t) (5 sin (t) + 4) = 0

分别求解每个部分

cos (t) = 0 or 5 sin (t) + 4 = 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

5 sin (t) + 4 = 0 : t = arcsin (- \frac{4}{5}) + 2 πn, t = π + arcsin (\frac{4}{5}) + 2 πn

5 sin (t) + 4 = 0

将

4

到右边

5 sin (t) + 4 = 0

两边减去

4

5 sin (t) + 4 - 4 = 0 - 4

化简

5 sin (t) = - 4

5 sin (t) = - 4

两边除以

5

5 sin (t) = - 4

两边除以

5

\frac{5 sin ( t )}{5} = \frac{- 4}{5}

化简

sin (t) = - \frac{4}{5}

sin (t) = - \frac{4}{5}

使用反三角函数性质

sin (t) = - \frac{4}{5}

sin (t) = - \frac{4}{5}

的通解

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

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

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

合并所有解

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

以小数形式表示解

t = \frac{π}{2} + 2 πn, t = \frac{3 π}{2} + 2 πn, t = - 0.92729 \dots + 2 πn, t = π + 0.92729 \dots + 2 πn

作图

流行的例子

cos(x+25)sec(65-x)=1

cos (x + 2 5^{\circ}) sec (6 5^{\circ} - x) = 1

tan^2(x)-5tan(x)-9=0

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arctan(t)= pi/4

arctan (t) = \frac{π}{4}

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sec(θ)(sec(θ)-1)=2

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