Trigonometry Examples

$2 \cos (θ) - 3 = 5 \cos (θ) - 5$

Step 1

Move all terms containing $\cos (θ)$ to the left side of the equation.

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Step 1.1

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

$2 \cos (θ) - 3 - 5 \cos (θ) = - 5$

Step 1.2

Subtract $5 \cos (θ)$ from $2 \cos (θ)$ .

$- 3 \cos (θ) - 3 = - 5$

Step 2

Move all terms not containing $\cos (θ)$ to the right side of the equation.

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Step 2.1

Add $3$ to both sides of the equation.

$- 3 \cos (θ) = - 5 + 3$

Step 2.2

Add $- 5$ and $3$ .

$- 3 \cos (θ) = - 2$

Step 3

Divide each term in $- 3 \cos (θ) = - 2$ by $- 3$ and simplify.

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Step 3.1

Divide each term in $- 3 \cos (θ) = - 2$ by $- 3$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 3$ .

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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.

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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.

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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 $\cos (θ)$ .

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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 $n$