What is the general solution for sin(θ)cos(3θ)+cos(θ)sin(3θ)=0 ?

The general solution for sin(θ)cos(3θ)+cos(θ)sin(3θ)=0 is θ=(pin)/2 ,θ= pi/4+(pin)/2

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sin(θ)cos(3θ)+cos(θ)sin(3θ)=0

sin (θ) cos (3 θ) + cos (θ) sin (3 θ) = 0

θ = \frac{πn}{2}, θ = \frac{π}{4} + \frac{πn}{2}

Degrees

θ = 0^{\circ} + 9 0^{\circ} n, θ = 4 5^{\circ} + 9 0^{\circ} n

Solution steps

sin (θ) cos (3 θ) + cos (θ) sin (3 θ) = 0

Rewrite using trig identities

sin (θ) cos (3 θ) + cos (θ) sin (3 θ)

Use the Angle Sum identity:

sin (s) cos (t) + cos (s) sin (t) = sin (s + t)

= sin (θ + 3 θ)

sin (θ + 3 θ) = 0

General solutions for

sin (θ + 3 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 + 3 θ = 0 + 2 πn, θ + 3 θ = π + 2 πn

θ + 3 θ = 0 + 2 πn, θ + 3 θ = π + 2 πn

Solve

θ + 3 θ = 0 + 2 πn : θ = \frac{πn}{2}

θ + 3 θ = 0 + 2 πn

Add similar elements:

θ + 3 θ = 4 θ

4 θ = 0 + 2 πn

0 + 2 πn = 2 πn

4 θ = 2 πn

Divide both sides by

4

4 θ = 2 πn

Divide both sides by

4

\frac{4 θ}{4} = \frac{2 πn}{4}

Simplify

θ = \frac{πn}{2}

θ = \frac{πn}{2}

Solve

θ + 3 θ = π + 2 πn : θ = \frac{π}{4} + \frac{πn}{2}

θ + 3 θ = π + 2 πn

Add similar elements:

θ + 3 θ = 4 θ

4 θ = π + 2 πn

Divide both sides by

4

4 θ = π + 2 πn

Divide both sides by

4

\frac{4 θ}{4} = \frac{π}{4} + \frac{2 πn}{4}

Simplify

θ = \frac{π}{4} + \frac{πn}{2}

θ = \frac{π}{4} + \frac{πn}{2}

θ = \frac{πn}{2}, θ = \frac{π}{4} + \frac{πn}{2}

19 sin (x) + 9 = sin (x)

cos (x) = \frac{3}{7}

tan (\frac{x}{4}) + 3 = 0

4 sin^{2} (x) tan (x) = tan (x)

cos (- x) = - cos (x)

What is the general solution for sin(θ)cos(3θ)+cos(θ)sin(3θ)=0 ?
The general solution for sin(θ)cos(3θ)+cos(θ)sin(3θ)=0 is θ=(pin)/2 ,θ= pi/4+(pin)/2