Solve for C in Degrees -5cos(C)-1=2cos(C)+3

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Trigonometry

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

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

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

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

Subtract $2 \cos (C)$ from $- 5 \cos (C)$ .

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

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

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Add $1$ to both sides of the equation.

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

Add $3$ and $1$ .

$- 7 \cos (C) = 4$

Divide each term in $- 7 \cos (C) = 4$ by $- 7$ and simplify.

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Divide each term in $- 7 \cos (C) = 4$ by $- 7$ .

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

Simplify the left side.

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Cancel the common factor of $- 7$ .

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Cancel the common factor.

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

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

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

Simplify the right side.

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Move the negative in front of the fraction.

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

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

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

Simplify the right side.

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

$C = 124.84990457$

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.

$C = 360 - 124.84990457$

Subtract $124.84990457$ from $360$ .

$C = 235.15009542$

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

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The period of the function can be calculated using $\frac{360}{| b |}$ .

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

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

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

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

$\frac{360}{1}$

Divide $360$ by $1$ .

$360$

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 $n$

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