Trigonometry Examples

$\sin (θ) = \frac{8}{10}$

Step 1

Cancel the common factor of $8$ and $10$ .

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Step 1.1

Factor $2$ out of $8$ .

$\sin (θ) = \frac{2 (4)}{10}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $2$ out of $10$ .

$\sin (θ) = \frac{2 \cdot 4}{2 \cdot 5}$

Step 1.2.2

Cancel the common factor.

$\sin (θ) = \frac{2 \cdot 4}{2 \cdot 5}$

Step 1.2.3

Rewrite the expression.

$\sin (θ) = \frac{4}{5}$

Step 2

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

$θ = \arcsin (\frac{4}{5})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arcsin (\frac{4}{5})$ .

$θ = 53.13010235$

Step 4

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the second quadrant.

$θ = 180 - 53.13010235$

Step 5

Subtract $53.13010235$ from $180$ .

$θ = 126.86989764$

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

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

$θ = 53.13010235 + 360 n, 126.86989764 + 360 n$ , for any integer $n$