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Popular Trigonometry >

16sin(t)cos(t)=4sin(t)

Solution

16 sin (t) cos (t) = 4 sin (t)

Solution

t = 2 πn, t = π + 2 πn, t = 1.31811 \dots + 2 πn, t = 2 π - 1.31811 \dots + 2 πn

+1

Degrees

t = 0^{\circ} + 36 0^{\circ} n, t = 18 0^{\circ} + 36 0^{\circ} n, t = 75.52248 \dots^{\circ} + 36 0^{\circ} n, t = 284.47751 \dots^{\circ} + 36 0^{\circ} n

Solution steps

16 sin (t) cos (t) = 4 sin (t)

Subtract

4 sin (t)

from both sides

16 sin (t) cos (t) - 4 sin (t) = 0

Factor

16 sin (t) cos (t) - 4 sin (t) : 4 sin (t) (4 cos (t) - 1)

16 sin (t) cos (t) - 4 sin (t)

Rewrite

16

as

4 \cdot 4

= 4 \cdot 4 sin (t) cos (t) - 4 sin (t)

Factor out common term

4 sin (t)

= 4 sin (t) (4 cos (t) - 1)

4 sin (t) (4 cos (t) - 1) = 0

Solving each part separately

sin (t) = 0 or 4 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

4 cos (t) - 1 = 0 : t = arccos (\frac{1}{4}) + 2 πn, t = 2 π - arccos (\frac{1}{4}) + 2 πn

4 cos (t) - 1 = 0

Move

1

to the right side

4 cos (t) - 1 = 0

Add

1

to both sides

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

Simplify

4 cos (t) = 1

4 cos (t) = 1

Divide both sides by

4

4 cos (t) = 1

Divide both sides by

4

\frac{4 cos ( t )}{4} = \frac{1}{4}

Simplify

cos (t) = \frac{1}{4}

cos (t) = \frac{1}{4}

Apply trig inverse properties

cos (t) = \frac{1}{4}

General solutions for

cos (t) = \frac{1}{4}

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

t = arccos (\frac{1}{4}) + 2 πn, t = 2 π - arccos (\frac{1}{4}) + 2 πn

t = arccos (\frac{1}{4}) + 2 πn, t = 2 π - arccos (\frac{1}{4}) + 2 πn

Combine all the solutions

t = 2 πn, t = π + 2 πn, t = arccos (\frac{1}{4}) + 2 πn, t = 2 π - arccos (\frac{1}{4}) + 2 πn

Show solutions in decimal form

t = 2 πn, t = π + 2 πn, t = 1.31811 \dots + 2 πn, t = 2 π - 1.31811 \dots + 2 πn

Graph

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