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Popular Trigonometry >

20sin(t)cos(t)=-4sin(t)

Solution

20 sin (t) cos (t) = - 4 sin (t)

Solution

t = 2 πn, t = π + 2 πn, t = 1.77215 \dots + 2 πn, t = - 1.77215 \dots + 2 πn

+1

Degrees

t = 0^{\circ} + 36 0^{\circ} n, t = 18 0^{\circ} + 36 0^{\circ} n, t = 101.53695 \dots^{\circ} + 36 0^{\circ} n, t = - 101.53695 \dots^{\circ} + 36 0^{\circ} n

Solution steps

20 sin (t) cos (t) = - 4 sin (t)

Subtract

- 4 sin (t)

from both sides

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

Factor

20 sin (t) cos (t) + 4 sin (t) : 4 sin (t) (5 cos (t) + 1)

20 sin (t) cos (t) + 4 sin (t)

Rewrite

20

as

5 \cdot 4

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

Factor out common term

4 sin (t)

= 4 sin (t) (5 cos (t) + 1)

4 sin (t) (5 cos (t) + 1) = 0

Solving each part separately

sin (t) = 0 or 5 cos (t) + 1 = 0

sin (t) = 0 : t = 2 πn, t = π + 2 πn

sin (t) = 0

General solutions for

sin (t) = 0

sin (x)

periodicity table with

2 πn

cycle:

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

t = 0 + 2 πn, t = π + 2 πn

t = 0 + 2 πn, t = π + 2 πn

Solve

t = 0 + 2 πn : t = 2 πn

t = 0 + 2 πn

0 + 2 πn = 2 πn

t = 2 πn

t = 2 πn, t = π + 2 πn

5 cos (t) + 1 = 0 : t = arccos (- \frac{1}{5}) + 2 πn, t = - arccos (- \frac{1}{5}) + 2 πn

5 cos (t) + 1 = 0

Move

1

to the right side

5 cos (t) + 1 = 0

Subtract

1

from both sides

5 cos (t) + 1 - 1 = 0 - 1

Simplify

5 cos (t) = - 1

5 cos (t) = - 1

Divide both sides by

5

5 cos (t) = - 1

Divide both sides by

5

\frac{5 cos ( t )}{5} = \frac{- 1}{5}

Simplify

cos (t) = - \frac{1}{5}

cos (t) = - \frac{1}{5}

Apply trig inverse properties

cos (t) = - \frac{1}{5}

General solutions for

cos (t) = - \frac{1}{5}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

t = arccos (- \frac{1}{5}) + 2 πn, t = - arccos (- \frac{1}{5}) + 2 πn

t = arccos (- \frac{1}{5}) + 2 πn, t = - arccos (- \frac{1}{5}) + 2 πn

Combine all the solutions

t = 2 πn, t = π + 2 πn, t = arccos (- \frac{1}{5}) + 2 πn, t = - arccos (- \frac{1}{5}) + 2 πn

Show solutions in decimal form

t = 2 πn, t = π + 2 πn, t = 1.77215 \dots + 2 πn, t = - 1.77215 \dots + 2 πn

Graph

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