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Popular Trigonometry >

3sin(θ)cos(θ)=sin(θ)

Solution

3 sin (θ) cos (θ) = sin (θ)

Solution

θ = 2 πn, θ = π + 2 πn, θ = 1.23095 \dots + 2 πn, θ = 2 π - 1.23095 \dots + 2 πn

+1

Degrees

θ = 0^{\circ} + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n, θ = 70.52877 \dots^{\circ} + 36 0^{\circ} n, θ = 289.47122 \dots^{\circ} + 36 0^{\circ} n

Solution steps

3 sin (θ) cos (θ) = sin (θ)

Subtract

sin (θ)

from both sides

3 sin (θ) cos (θ) - sin (θ) = 0

Factor

3 sin (θ) cos (θ) - sin (θ) : sin (θ) (3 cos (θ) - 1)

3 sin (θ) cos (θ) - sin (θ)

Factor out common term

sin (θ)

= sin (θ) (3 cos (θ) - 1)

sin (θ) (3 cos (θ) - 1) = 0

Solving each part separately

sin (θ) = 0 or 3 cos (θ) - 1 = 0

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

sin (θ) = 0

General solutions for

sin (θ) = 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}

θ = 0 + 2 πn, θ = π + 2 πn

θ = 0 + 2 πn, θ = π + 2 πn

Solve

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn, θ = π + 2 πn

3 cos (θ) - 1 = 0 : θ = arccos (\frac{1}{3}) + 2 πn, θ = 2 π - arccos (\frac{1}{3}) + 2 πn

3 cos (θ) - 1 = 0

Move

1

to the right side

3 cos (θ) - 1 = 0

Add

1

to both sides

3 cos (θ) - 1 + 1 = 0 + 1

Simplify

3 cos (θ) = 1

3 cos (θ) = 1

Divide both sides by

3

3 cos (θ) = 1

Divide both sides by

3

\frac{3 cos ( θ )}{3} = \frac{1}{3}

Simplify

cos (θ) = \frac{1}{3}

cos (θ) = \frac{1}{3}

Apply trig inverse properties

cos (θ) = \frac{1}{3}

General solutions for

cos (θ) = \frac{1}{3}

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

θ = arccos (\frac{1}{3}) + 2 πn, θ = 2 π - arccos (\frac{1}{3}) + 2 πn

θ = arccos (\frac{1}{3}) + 2 πn, θ = 2 π - arccos (\frac{1}{3}) + 2 πn

Combine all the solutions

θ = 2 πn, θ = π + 2 πn, θ = arccos (\frac{1}{3}) + 2 πn, θ = 2 π - arccos (\frac{1}{3}) + 2 πn

Show solutions in decimal form

θ = 2 πn, θ = π + 2 πn, θ = 1.23095 \dots + 2 πn, θ = 2 π - 1.23095 \dots + 2 πn

Graph

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