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

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

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sin(x)+sqrt(3)cos(x)=0

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

x = \frac{2 π}{3} + πn

Degrees

x = 12 0^{\circ} + 18 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Divide both sides by

cos (x), cos (x) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

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

tan (x) + 3 = 0

tan (x) + 3 = 0

Move

3

to the right side

tan (x) + 3 = 0

Subtract

3

from both sides

tan (x) + 3 - 3 = 0 - 3

Simplify

tan (x) = - 3

tan (x) = - 3

General solutions for

tan (x) = - 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}

x = \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

cos^{2} (x) + 2 cos (x) - 3 = 0

cos (θ) = \frac{3}{4}

sin (x) - 3 cos (x) = 0

cot (θ) = - 3

4 sin^{2} (x) - 2 = 0

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