Trigonometry Examples

5 cos (A) + 8 = 3 cos (A) + 6

$5 \cos (A) + 8 = 3 \cos (A) + 6$

Step 1

Move all terms containing

cos (A)

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

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Step 1.1

Subtract

3 cos (A)

$3 \cos (A)$ from both sides of the equation.

5 cos (A) + 8 - 3 cos (A) = 6

$5 \cos (A) + 8 - 3 \cos (A) = 6$

Step 1.2

Subtract

3 cos (A)

$3 \cos (A)$ from

5 cos (A)

$5 \cos (A)$ .

2 cos (A) + 8 = 6

$2 \cos (A) + 8 = 6$

2 cos (A) + 8 = 6

$2 \cos (A) + 8 = 6$

Step 2

Move all terms not containing

cos (A)

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

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Step 2.1

Subtract

8

$8$ from both sides of the equation.

2 cos (A) = 6 - 8

$2 \cos (A) = 6 - 8$

Step 2.2

Subtract

8

$8$ from

6

$6$ .

2 cos (A) = - 2

$2 \cos (A) = - 2$

2 cos (A) = - 2

$2 \cos (A) = - 2$

Step 3

Divide each term in

2 cos (A) = - 2

$2 \cos (A) = - 2$ by

2

$2$ and simplify.

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Step 3.1

Divide each term in

2 cos (A) = - 2

$2 \cos (A) = - 2$ by

2

$2$ .

\frac{2 cos (A)}{2} = \frac{- 2}{2}

$\frac{2 \cos (A)}{2} = \frac{- 2}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of

2

$2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 \cos (A)}{2} = \frac{- 2}{2}$

Step 3.2.1.2

Divide

$\cos (A)$ by

$1$ .

$\cos (A) = \frac{- 2}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide

$- 2$ by

$2$ .

$\cos (A) = - 1$

Step 4

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

$A$ from inside the cosine.

$A = \arccos (- 1)$

Step 5

Simplify the right side.

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Step 5.1

The exact value of

$\arccos (- 1)$ is

$180$ .

$A = 180$

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.

$A = 360 - 180$

Step 7

Subtract

$180$ from

$360$ .

$A = 180$

Step 8

Find the period of

$\cos (A)$ .

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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

$\cos (A)$ function is

$360$ so values will repeat every

$360$ degrees in both directions.

$A = 180 + 360 n$ , for any integer

$n$