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

The general solution for sin(θ)=sqrt(3)cos(θ) is θ= pi/3+pin

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sin(θ)=sqrt(3)cos(θ)

sin (θ) = 3 cos (θ)

θ = \frac{π}{3} + πn

Degrees

θ = 6 0^{\circ} + 18 0^{\circ} n

Solution steps

sin (θ) = 3 cos (θ)

Rewrite using trig identities

sin (θ) = 3 cos (θ)

Divide both sides by

cos (θ), cos (θ) \neq = 0

\frac{sin ( θ )}{cos ( θ )} = \frac{3 cos ( θ )}{cos ( θ )}

Simplify

\frac{sin ( θ )}{cos ( θ )} = 3

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

tan (θ) = 3

tan (θ) = 3

General solutions for

tan (θ) = 3

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

θ = \frac{π}{3} + πn

θ = \frac{π}{3} + πn

2 (cos^{2} (x) - sin^{2} (x)) = 1

- sin (θ) + cos (2 θ) = 1

arctan (2 x - 8) = - 1

- 2 cos (2 θ) - 5 = - 4 sin (θ) - 8

3 cos^{2} (θ) - cos (θ) = 0

What is the general solution for sin(θ)=sqrt(3)cos(θ) ?
The general solution for sin(θ)=sqrt(3)cos(θ) is θ= pi/3+pin