Trigonometry Examples

$2 \sin (θ) = - 1$

Step 1

Divide each term in $2 \sin (θ) = - 1$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 \sin (θ) = - 1$ by $2$ .

$\frac{2 \sin (θ)}{2} = \frac{- 1}{2}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{2 \sin (θ)}{2} = \frac{- 1}{2}$

Step 1.2.1.2

Divide $\sin (θ)$ by $1$ .

$\sin (θ) = \frac{- 1}{2}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Move the negative in front of the fraction.

$\sin (θ) = - \frac{1}{2}$

Step 2

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

$θ = \arcsin (- \frac{1}{2})$

Step 3

Simplify the right side.

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Step 3.1

The exact value of $\arcsin (- \frac{1}{2})$ is $- 30$ .

$θ = - 30$

Step 4

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $360$ , to find a reference angle. Next, add this reference angle to $180$ to find the solution in the third quadrant.

$θ = 360 + 30 + 180$

Step 5

Simplify the expression to find the second solution.

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Step 5.1

Subtract $360 °$ from $360 + 30 + 180 °$ .

$θ = 360 + 30 + 180 ° - 360 °$

Step 5.2

The resulting angle of $210 °$ is positive, less than $360 °$ , and coterminal with $360 + 30 + 180$ .

$θ = 210 °$

Step 6

Find the period of $\sin (θ)$ .

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

$\frac{360}{1}$

Step 6.4

Divide $360$ by $1$ .

$360$

Step 7

Add $360$ to every negative angle to get positive angles.

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Step 7.1

Add $360$ to $- 30$ to find the positive angle.

$- 30 + 360$

Step 7.2

Subtract $30$ from $360$ .

$330$

Step 7.3

List the new angles.

$θ = 330$

Step 8

The period of the $\sin (θ)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$θ = 210 + 360 n, 330 + 360 n$ , for any integer $n$