Trigonometry Examples

$30 \cos (30 x) + 14 = - 16$

Step 1

Move all terms not containing $\cos (30 x)$ to the right side of the equation.

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Step 1.1

Subtract $14$ from both sides of the equation.

$30 \cos (30 x) = - 16 - 14$

Step 1.2

Subtract $14$ from $- 16$ .

$30 \cos (30 x) = - 30$

Step 2

Divide each term in $30 \cos (30 x) = - 30$ by $30$ and simplify.

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Step 2.1

Divide each term in $30 \cos (30 x) = - 30$ by $30$ .

$\frac{30 \cos (30 x)}{30} = \frac{- 30}{30}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $30$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{30 \cos (30 x)}{30} = \frac{- 30}{30}$

Step 2.2.1.2

Divide $\cos (30 x)$ by $1$ .

$\cos (30 x) = \frac{- 30}{30}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $- 30$ by $30$ .

$\cos (30 x) = - 1$

Step 3

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

$30 x = \arccos (- 1)$

Step 4

Simplify the right side.

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Step 4.1

The exact value of $\arccos (- 1)$ is $π$ .

$30 x = π$

Step 5

Divide each term in $30 x = π$ by $30$ and simplify.

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Step 5.1

Divide each term in $30 x = π$ by $30$ .

$\frac{30 x}{30} = \frac{π}{30}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $30$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{30 x}{30} = \frac{π}{30}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{30}$

Step 6

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

$30 x = 2 π - π$

Step 7

Solve for $x$ .

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Step 7.1

Subtract $π$ from $2 π$ .

$30 x = π$

Step 7.2

Divide each term in $30 x = π$ by $30$ and simplify.

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Step 7.2.1

Divide each term in $30 x = π$ by $30$ .

$\frac{30 x}{30} = \frac{π}{30}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $30$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{30 x}{30} = \frac{π}{30}$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{30}$

Step 8

Find the period of $\cos (30 x)$ .

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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 $30$ in the formula for period.

$\frac{2 π}{| 30 |}$

Step 8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $30$ is $30$ .

$\frac{2 π}{30}$

Step 8.4

Cancel the common factor of $2$ and $30$ .

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Step 8.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{30}$

Step 8.4.2

Cancel the common factors.

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Step 8.4.2.1

Factor $2$ out of $30$ .

$\frac{2 π}{2 \cdot 15}$

Step 8.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 15}$

Step 8.4.2.3

Rewrite the expression.

$\frac{π}{15}$

Step 9

The period of the $\cos (30 x)$ function is $\frac{π}{15}$ so values will repeat every $\frac{π}{15}$ radians in both directions.

$x = \frac{π}{30} + \frac{π n}{15}$ , for any integer $n$