Trigonometry Examples

$1 = 8 \cos (x + 1) - 3$

Step 1

Rewrite the equation as $8 \cos (x + 1) - 3 = 1$ .

$8 \cos (x + 1) - 3 = 1$

Step 2

Move all terms not containing $\cos (x + 1)$ to the right side of the equation.

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Step 2.1

Add $3$ to both sides of the equation.

$8 \cos (x + 1) = 1 + 3$

Step 2.2

Add $1$ and $3$ .

$8 \cos (x + 1) = 4$

Step 3

Divide each term in $8 \cos (x + 1) = 4$ by $8$ and simplify.

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Step 3.1

Divide each term in $8 \cos (x + 1) = 4$ by $8$ .

$\frac{8 \cos (x + 1)}{8} = \frac{4}{8}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $8$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{8 \cos (x + 1)}{8} = \frac{4}{8}$

Step 3.2.1.2

Divide $\cos (x + 1)$ by $1$ .

$\cos (x + 1) = \frac{4}{8}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $4$ and $8$ .

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Step 3.3.1.1

Factor $4$ out of $4$ .

$\cos (x + 1) = \frac{4 (1)}{8}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $4$ out of $8$ .

$\cos (x + 1) = \frac{4 \cdot 1}{4 \cdot 2}$

Step 3.3.1.2.2

Cancel the common factor.

$\cos (x + 1) = \frac{4 \cdot 1}{4 \cdot 2}$

Step 3.3.1.2.3

Rewrite the expression.

$\cos (x + 1) = \frac{1}{2}$

Step 4

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

$x + 1 = \arccos (\frac{1}{2})$

Step 5

Simplify the right side.

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Step 5.1

The exact value of $\arccos (\frac{1}{2})$ is $\frac{π}{3}$ .

$x + 1 = \frac{π}{3}$

Step 6

Subtract $1$ from both sides of the equation.

$x = \frac{π}{3} - 1$

Step 7

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.

$x + 1 = 2 π - \frac{π}{3}$

Step 8

Solve for $x$ .

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Step 8.1

Simplify $2 π - \frac{π}{3}$ .

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Step 8.1.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x + 1 = 2 π \cdot \frac{3}{3} - \frac{π}{3}$

Step 8.1.2

Combine fractions.

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Step 8.1.2.1

Combine $2 π$ and $\frac{3}{3}$ .

$x + 1 = \frac{2 π \cdot 3}{3} - \frac{π}{3}$

Step 8.1.2.2

Combine the numerators over the common denominator.

$x + 1 = \frac{2 π \cdot 3 - π}{3}$

Step 8.1.3

Simplify the numerator.

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Step 8.1.3.1

Multiply $3$ by $2$ .

$x + 1 = \frac{6 π - π}{3}$

Step 8.1.3.2

Subtract $π$ from $6 π$ .

$x + 1 = \frac{5 π}{3}$

Step 8.2

Subtract $1$ from both sides of the equation.

$x = \frac{5 π}{3} - 1$

Step 9

Find the period of $\cos (x + 1)$ .

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Step 9.1

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

$\frac{2 π}{| b |}$

Step 9.2

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

$\frac{2 π}{| 1 |}$

Step 9.3

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

$\frac{2 π}{1}$

Step 9.4

Divide $2 π$ by $1$ .

$2 π$

Step 10

The period of the $\cos (x + 1)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{π}{3} - 1 + 2 π n, \frac{5 π}{3} - 1 + 2 π n$ , for any integer $n$