Trigonometry Examples

- 5 cos (C) - 1 = 2 cos (C) + 3

$- 5 \cos (C) - 1 = 2 \cos (C) + 3$

Step 1

Move all terms containing

cos (C)

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

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Step 1.1

Subtract

2 cos (C)

$2 \cos (C)$ from both sides of the equation.

- 5 cos (C) - 1 - 2 cos (C) = 3

$- 5 \cos (C) - 1 - 2 \cos (C) = 3$

Step 1.2

Subtract

2 cos (C)

$2 \cos (C)$ from

- 5 cos (C)

$- 5 \cos (C)$ .

- 7 cos (C) - 1 = 3

$- 7 \cos (C) - 1 = 3$

- 7 cos (C) - 1 = 3

$- 7 \cos (C) - 1 = 3$

Step 2

Move all terms not containing

cos (C)

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

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Step 2.1

Add

1

$1$ to both sides of the equation.

- 7 cos (C) = 3 + 1

$- 7 \cos (C) = 3 + 1$

Step 2.2

Add

3

$3$ and

1

$1$ .

- 7 cos (C) = 4

$- 7 \cos (C) = 4$

- 7 cos (C) = 4

$- 7 \cos (C) = 4$

Step 3

Divide each term in

- 7 cos (C) = 4

$- 7 \cos (C) = 4$ by

- 7

$- 7$ and simplify.

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Step 3.1

Divide each term in

- 7 cos (C) = 4

$- 7 \cos (C) = 4$ by

- 7

$- 7$ .

\frac{- 7 cos (C)}{- 7} = \frac{4}{- 7}

$\frac{- 7 \cos (C)}{- 7} = \frac{4}{- 7}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of

- 7

$- 7$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 7 \cos (C)}{- 7} = \frac{4}{- 7}$

Step 3.2.1.2

Divide

$\cos (C)$ by

$1$ .

$\cos (C) = \frac{4}{- 7}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

$\cos (C) = - \frac{4}{7}$

Step 4

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

$C$ from inside the cosine.

$C = \arccos (- \frac{4}{7})$

Step 5

Simplify the right side.

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Step 5.1

Evaluate

$\arccos (- \frac{4}{7})$ .

$C = 124.84990457$

Step 6

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from

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

$C = 360 - 124.84990457$

Step 7

Subtract

$124.84990457$ from

$360$ .

$C = 235.15009542$

Step 8

Find the period of

$\cos (C)$ .

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Step 8.1

The period of the function can be calculated using

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

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

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

$\cos (C)$ function is

$360$ so values will repeat every

$360$ degrees in both directions.

$C = 124.84990457 + 360 n, 235.15009542 + 360 n$ , for any integer

$n$