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

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

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sin(θ)+sqrt(3)cos(θ)=0

sin (θ) + 3 cos (θ) = 0

θ = \frac{2 π}{3} + πn

Degrees

θ = 12 0^{\circ} + 18 0^{\circ} n

Solution steps

sin (θ) + 3 cos (θ) = 0

Rewrite using trig identities

sin (θ) + 3 cos (θ) = 0

Divide both sides by

cos (θ), cos (θ) \neq = 0

\frac{sin ( θ ) + 3 cos ( θ )}{cos ( θ )} = \frac{0}{cos ( θ )}

Simplify

\frac{sin ( θ )}{cos ( θ )} + 3 = 0

Use the basic trigonometric identity:

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

tan (θ) + 3 = 0

tan (θ) + 3 = 0

Move

3

to the right side

tan (θ) + 3 = 0

Subtract

3

from both sides

tan (θ) + 3 - 3 = 0 - 3

Simplify

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{2 π}{3} + πn

θ = \frac{2 π}{3} + πn

cos (x) + 2 sin (x) cos (x) = 0

3 tan (x) + 3 = 0

(2 sin^{2} (x) - 1) (3 tan^{2} (x) - 1) = 0

sec (x) - 1 = 0

cos (x) + 2 sin (x) = 2

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