Trigonometry Examples

2 cos (θ) - 3 = 5 cos (θ) - 5

$2 \cos (θ) - 3 = 5 \cos (θ) - 5$

Step 1

Move all terms containing

cos (θ)

$\cos (θ)$ to the left side of the equation.

Tap for more steps...

Step 1.1

Subtract

5 cos (θ)

$5 \cos (θ)$ from both sides of the equation.

2 cos (θ) - 3 - 5 cos (θ) = - 5

$2 \cos (θ) - 3 - 5 \cos (θ) = - 5$

Step 1.2

Subtract

5 cos (θ)

$5 \cos (θ)$ from

2 cos (θ)

$2 \cos (θ)$ .

- 3 cos (θ) - 3 = - 5

$- 3 \cos (θ) - 3 = - 5$

- 3 cos (θ) - 3 = - 5

$- 3 \cos (θ) - 3 = - 5$

Step 2

Move all terms not containing

cos (θ)

$\cos (θ)$ to the right side of the equation.

Tap for more steps...

Step 2.1

Add

3

$3$ to both sides of the equation.

- 3 cos (θ) = - 5 + 3

$- 3 \cos (θ) = - 5 + 3$

Step 2.2

Add

- 5

$- 5$ and

3

$3$ .

- 3 cos (θ) = - 2

$- 3 \cos (θ) = - 2$

- 3 cos (θ) = - 2

$- 3 \cos (θ) = - 2$

Step 3

Divide each term in

- 3 cos (θ) = - 2

$- 3 \cos (θ) = - 2$ by

- 3

$- 3$ and simplify.

Tap for more steps...

Step 3.1

Divide each term in

- 3 cos (θ) = - 2

$- 3 \cos (θ) = - 2$ by

- 3

$- 3$ .

\frac{- 3 cos (θ)}{- 3} = \frac{- 2}{- 3}

$\frac{- 3 \cos (θ)}{- 3} = \frac{- 2}{- 3}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Cancel the common factor of

- 3

$- 3$ .

Tap for more steps...

Step 3.2.1.1

Cancel the common factor.

$\frac{- 3 \cos (θ)}{- 3} = \frac{- 2}{- 3}$

Step 3.2.1.2

Divide

$\cos (θ)$ by

$1$ .

$\cos (θ) = \frac{- 2}{- 3}$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Dividing two negative values results in a positive value.

$\cos (θ) = \frac{2}{3}$

Step 4

Take the inverse cosine of both sides of the equation to extract

$θ$ from inside the cosine.

$θ = \arccos (\frac{2}{3})$

Step 5

Simplify the right side.

Tap for more steps...

Step 5.1

Evaluate

$\arccos (\frac{2}{3})$ .

$θ = 48.1896851$

Step 6

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from

$360$ to find the solution in the fourth quadrant.

$θ = 360 - 48.1896851$

Step 7

Subtract

$48.1896851$ from

$360$ .

$θ = 311.81031489$

Step 8

Find the period of

$\cos (θ)$ .

Tap for more steps...

Step 8.1

The period of the function can be calculated using

$\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 8.2

Replace

$b$ with

$1$ in the formula for period.

$\frac{360}{| 1 |}$

Step 8.3

The absolute value is the distance between a number and zero. The distance between

$0$ and

$1$ is

$1$ .

$\frac{360}{1}$

Step 8.4

Divide

$360$ by

$1$ .

$360$

Step 9

The period of the

$\cos (θ)$ function is

$360$ so values will repeat every

$360$ degrees in both directions.

$θ = 48.1896851 + 360 n, 311.81031489 + 360 n$ , for any integer

$n$