Trigonometry Examples

$86.49 = 123.85 - 122.64 \cos (c)$

Step 1

Rewrite the equation as $123.85 - 122.64 \cos (c) = 86.49$ .

$123.85 - 122.64 \cos (c) = 86.49$

Step 2

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

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Step 2.1

Subtract $123.85$ from both sides of the equation.

$- 122.64 \cos (c) = 86.49 - 123.85$

Step 2.2

Subtract $123.85$ from $86.49$ .

$- 122.64 \cos (c) = - 37.36$

Step 3

Divide each term in $- 122.64 \cos (c) = - 37.36$ by $- 122.64$ and simplify.

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Step 3.1

Divide each term in $- 122.64 \cos (c) = - 37.36$ by $- 122.64$ .

$\frac{- 122.64 \cos (c)}{- 122.64} = \frac{- 37.36}{- 122.64}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 122.64$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 122.64 \cos (c)}{- 122.64} = \frac{- 37.36}{- 122.64}$

Step 3.2.1.2

Divide $\cos (c)$ by $1$ .

$\cos (c) = \frac{- 37.36}{- 122.64}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $- 37.36$ by $- 122.64$ .

$\cos (c) = 0.30463144$

Step 4

Take the inverse cosine of both sides of the equation to extract $c$ from inside the cosine.

$c = \arccos (0.30463144)$

Step 5

Simplify the right side.

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Step 5.1

Evaluate $\arccos (0.30463144)$ .

$c = 1.26124487$

Step 6

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$c = 2 (3.14159265) - 1.26124487$

Step 7

Simplify $2 (3.14159265) - 1.26124487$ .

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Step 7.1

Multiply $2$ by $3.14159265$ .

$c = 6.2831853 - 1.26124487$

Step 7.2

Subtract $1.26124487$ from $6.2831853$ .

$c = 5.02194043$

Step 8

Find the period of $\cos (c)$ .

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Step 8.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 8.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 8.4

Divide $2 π$ by $1$ .

$2 π$

Step 9

The period of the $\cos (c)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$c = 1.26124487 + 2 π n, 5.02194043 + 2 π n$ , for any integer $n$