What is the general solution for 5cos(θ)=-5sqrt(3)sin(-θ) ?

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

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Popular >

5cos(θ)=-5sqrt(3)sin(-θ)

Solution

5 cos (θ) = - 53 sin (- θ)

Solution

θ = \frac{π}{6} + πn

Degrees

θ = 3 0^{\circ} + 18 0^{\circ} n

Solution steps

5 cos (θ) = - 53 sin (- θ)

Rewrite using trig identities

5 cos (θ) = - 53 sin (- θ)

5 cos (θ) = - 53 (- sin (θ))

Simplify

- 53 (- sin (θ)) : 53 sin (θ)

- 53 (- sin (θ))

Apply rule

- (- a) = a

= 53 sin (θ)

5 cos (θ) = 53 sin (θ)

5 cos (θ) = 53 sin (θ)

Subtract

53 sin (θ)

from both sides

5 cos (θ) - 53 sin (θ) = 0

Rewrite using trig identities

5 cos (θ) - 53 sin (θ) = 0

Divide both sides by

cos (θ), cos (θ) \neq = 0

\frac{5 cos ( θ ) - 5 3 sin ( θ )}{cos ( θ )} = \frac{0}{cos ( θ )}

Simplify

5 - \frac{5 3 sin ( θ )}{cos ( θ )} = 0

Use the basic trigonometric identity:

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

5 - 53 tan (θ) = 0

5 - 53 tan (θ) = 0

Move

5

to the right side

5 - 53 tan (θ) = 0

Subtract

5

from both sides

5 - 53 tan (θ) - 5 = 0 - 5

Simplify

- 53 tan (θ) = - 5

- 53 tan (θ) = - 5

Divide both sides by

- 53

- 53 tan (θ) = - 5

Divide both sides by

- 53

\frac{- 5 3 tan ( θ )}{- 5 3} = \frac{- 5}{- 5 3}

Simplify

\frac{- 5 3 tan ( θ )}{- 5 3} = \frac{- 5}{- 5 3}

Simplify

\frac{- 5 3 tan ( θ )}{- 5 3} : tan (θ)

\frac{- 5 3 tan ( θ )}{- 5 3}

Apply the fraction rule:

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

= \frac{5 3 tan ( θ )}{5 3}

Divide the numbers:

\frac{5}{5} = 1

= \frac{3 tan ( θ )}{3}

Cancel the common factor:

3

= tan (θ)

Simplify

\frac{- 5}{- 5 3} : \frac{3}{3}

\frac{- 5}{- 5 3}

Apply the fraction rule:

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

= \frac{5}{5 3}

Divide the numbers:

\frac{5}{5} = 1

= \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{π}{6} + πn

θ = \frac{π}{6} + πn

Popular Examples

sin(x)=-15/17

sin (x) = - \frac{15}{17}

tan(θ)=7

tan (θ) = 7

0=sin(x^2)

0 = sin (x^{2})

3cos(2θ)=0

3 cos (2 θ) = 0

4cos(2x)+2sin(x)=1,0<= x<= 360

4 cos (2 x) + 2 sin (x) = 1, 0^{\circ} \leq x \leq 36 0^{\circ}

Frequently Asked Questions (FAQ)

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