Trigonometry Examples

$\sin (5 x) - 16 = - 17$

Step 1

Move all terms not containing $\sin (5 x)$ to the right side of the equation.

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Step 1.1

Add $16$ to both sides of the equation.

$\sin (5 x) = - 17 + 16$

Step 1.2

Add $- 17$ and $16$ .

$\sin (5 x) = - 1$

Step 2

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

$5 x = \arcsin (- 1)$

Step 3

Simplify the right side.

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Step 3.1

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

$5 x = - \frac{π}{2}$

Step 4

Divide each term in $5 x = - \frac{π}{2}$ by $5$ and simplify.

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Step 4.1

Divide each term in $5 x = - \frac{π}{2}$ by $5$ .

$\frac{5 x}{5} = \frac{- \frac{π}{2}}{5}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $5$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- \frac{π}{2}}{5}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{π}{2}}{5}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{π}{2} \cdot \frac{1}{5}$

Step 4.3.2

Multiply $- \frac{π}{2} \cdot \frac{1}{5}$ .

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Step 4.3.2.1

Multiply $\frac{1}{5}$ by $\frac{π}{2}$ .

$x = - \frac{π}{5 \cdot 2}$

Step 4.3.2.2

Multiply $5$ by $2$ .

$x = - \frac{π}{10}$

Step 5

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$5 x = 2 π + \frac{π}{2} + π$

Step 6

Simplify the expression to find the second solution.

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Step 6.1

Subtract $2 π$ from $2 π + \frac{π}{2} + π$ .

$5 x = 2 π + \frac{π}{2} + π - 2 π$

Step 6.2

The resulting angle of $\frac{3 π}{2}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{2} + π$ .

$5 x = \frac{3 π}{2}$

Step 6.3

Divide each term in $5 x = \frac{3 π}{2}$ by $5$ and simplify.

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Step 6.3.1

Divide each term in $5 x = \frac{3 π}{2}$ by $5$ .

$\frac{5 x}{5} = \frac{\frac{3 π}{2}}{5}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $5$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{\frac{3 π}{2}}{5}$

Step 6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{5}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{5}$

Step 6.3.3.2

Multiply $\frac{3 π}{2} \cdot \frac{1}{5}$ .

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Step 6.3.3.2.1

Multiply $\frac{3 π}{2}$ by $\frac{1}{5}$ .

$x = \frac{3 π}{2 \cdot 5}$

Step 6.3.3.2.2

Multiply $2$ by $5$ .

$x = \frac{3 π}{10}$

Step 7

Find the period of $\sin (5 x)$ .

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Step 7.1

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

$\frac{2 π}{| b |}$

Step 7.2

Replace $b$ with $5$ in the formula for period.

$\frac{2 π}{| 5 |}$

Step 7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$\frac{2 π}{5}$

Step 8

Add $\frac{2 π}{5}$ to every negative angle to get positive angles.

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Step 8.1

Add $\frac{2 π}{5}$ to $- \frac{π}{10}$ to find the positive angle.

$- \frac{π}{10} + \frac{2 π}{5}$

Step 8.2

To write $\frac{2 π}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2 π}{5} \cdot \frac{2}{2} - \frac{π}{10}$

Step 8.3

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.3.1

Multiply $\frac{2 π}{5}$ by $\frac{2}{2}$ .

$\frac{2 π \cdot 2}{5 \cdot 2} - \frac{π}{10}$

Step 8.3.2

Multiply $5$ by $2$ .

$\frac{2 π \cdot 2}{10} - \frac{π}{10}$

Step 8.4

Combine the numerators over the common denominator.

$\frac{2 π \cdot 2 - π}{10}$

Step 8.5

Simplify the numerator.

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Step 8.5.1

Multiply $2$ by $2$ .

$\frac{4 π - π}{10}$

Step 8.5.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{10}$

Step 8.6

List the new angles.

$x = \frac{3 π}{10}$

Step 9

The period of the $\sin (5 x)$ function is $\frac{2 π}{5}$ so values will repeat every $\frac{2 π}{5}$ radians in both directions.

$x = \frac{3 π}{10} + \frac{2 π n}{5}, \frac{3 π}{10} + \frac{2 π n}{5}$ , for any integer $n$

Step 10

Consolidate the answers.

$x = \frac{3 π}{10} + \frac{2 π n}{5}$ , for any integer $n$