What is the general solution for tan(θ)sin(θ)=3sin(θ) ?

The general solution for tan(θ)sin(θ)=3sin(θ) is θ=2pin,θ=pi+2pin,θ=1.24904…+pin

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tan(θ)sin(θ)=3sin(θ)

tan (θ) sin (θ) = 3 sin (θ)

θ = 2 πn, θ = π + 2 πn, θ = 1.24904 \dots + πn

Degrees

θ = 0^{\circ} + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n, θ = 71.56505 \dots^{\circ} + 18 0^{\circ} n

Solution steps

tan (θ) sin (θ) = 3 sin (θ)

Subtract

3 sin (θ)

from both sides

tan (θ) sin (θ) - 3 sin (θ) = 0

Factor

tan (θ) sin (θ) - 3 sin (θ) : sin (θ) (tan (θ) - 3)

tan (θ) sin (θ) - 3 sin (θ)

Factor out common term

sin (θ)

= sin (θ) (tan (θ) - 3)

sin (θ) (tan (θ) - 3) = 0

Solving each part separately

sin (θ) = 0 or tan (θ) - 3 = 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

tan (θ) - 3 = 0 : θ = arctan (3) + πn

tan (θ) - 3 = 0

Move

3

to the right side

tan (θ) - 3 = 0

Add

3

to both sides

tan (θ) - 3 + 3 = 0 + 3

Simplify

tan (θ) = 3

tan (θ) = 3

Apply trig inverse properties

tan (θ) = 3

General solutions for

tan (θ) = 3

tan (x) = a \Rightarrow x = arctan (a) + πn

θ = arctan (3) + πn

θ = arctan (3) + πn

Combine all the solutions

θ = 2 πn, θ = π + 2 πn, θ = arctan (3) + πn

Show solutions in decimal form

θ = 2 πn, θ = π + 2 πn, θ = 1.24904 \dots + πn

sin (2 x) = 1 - sin (2 x)

3 tan (c) + 8 = 0

cos^{2} (x) - 2 cos (x) - 2 = 0

tan (x) = \frac{14.9}{13.3}

3 sin^{2} (x) + sin (x) - 4 = 0, 0 \leq x \leq 2 π

What is the general solution for tan(θ)sin(θ)=3sin(θ) ?
The general solution for tan(θ)sin(θ)=3sin(θ) is θ=2pin,θ=pi+2pin,θ=1.24904…+pin