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

The general solution for sqrt(3)sin(θ)+cos(θ)=0 is θ=(5pi)/6+pin

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Popular >

sqrt(3)sin(θ)+cos(θ)=0

Solution

3 sin (θ) + cos (θ) = 0

Solution

θ = \frac{5 π}{6} + πn

Degrees

θ = 15 0^{\circ} + 18 0^{\circ} n

Solution steps

3 sin (θ) + cos (θ) = 0

Rewrite using trig identities

3 sin (θ) + cos (θ) = 0

Divide both sides by

cos (θ), cos (θ) \neq = 0

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

Simplify

\frac{3 sin ( θ )}{cos ( θ )} + 1 = 0

Use the basic trigonometric identity:

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

3 tan (θ) + 1 = 0

3 tan (θ) + 1 = 0

Move

1

to the right side

3 tan (θ) + 1 = 0

Subtract

1

from both sides

3 tan (θ) + 1 - 1 = 0 - 1

Simplify

3 tan (θ) = - 1

3 tan (θ) = - 1

Divide both sides by

3

3 tan (θ) = - 1

Divide both sides by

3

\frac{3 tan ( θ )}{3} = \frac{- 1}{3}

Simplify

\frac{3 tan ( θ )}{3} = \frac{- 1}{3}

Simplify

\frac{3 tan ( θ )}{3} : tan (θ)

\frac{3 tan ( θ )}{3}

Cancel the common factor:

3

= tan (θ)

Simplify

\frac{- 1}{3} : - \frac{3}{3}

\frac{- 1}{3}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1}{3}

Rationalize

- \frac{1}{3} : - \frac{3}{3}

- \frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= - \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= - \frac{3}{3}

= - \frac{3}{3}

tan (θ) = - \frac{3}{3}

tan (θ) = - \frac{3}{3}

tan (θ) = - \frac{3}{3}

General solutions for

tan (θ) = - \frac{3}{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{5 π}{6} + πn

θ = \frac{5 π}{6} + πn

Popular Examples

cos^2(x)= 1/4

cos^{2} (x) = \frac{1}{4}

sin(x)+sqrt(2)=-sin(x)

sin (x) + 2 = - sin (x)

cos(x)= 12/13

cos (x) = \frac{12}{13}

cos(x)(2sin(x)+1)=0

cos (x) (2 sin (x) + 1) = 0

cos(y)=0

cos (y) = 0

Frequently Asked Questions (FAQ)

What is the general solution for sqrt(3)sin(θ)+cos(θ)=0 ?
The general solution for sqrt(3)sin(θ)+cos(θ)=0 is θ=(5pi)/6+pin