Trigonometry Examples

$\sin (B) = \frac{- 5}{8}$

Step 1

Move the negative in front of the fraction.

$\sin (B) = - \frac{5}{8}$

Step 2

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

$B = \arcsin (- \frac{5}{8})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arcsin (- \frac{5}{8})$ .

$B = - 0.67513153$

Step 4

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.

$B = 2 (3.14159265) + 0.67513153 + 3.14159265$

Step 5

Simplify the expression to find the second solution.

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Step 5.1

Subtract $2 π$ from $2 (3.14159265) + 0.67513153 + 3.14159265$ .

$B = 2 (3.14159265) + 0.67513153 + 3.14159265 - 2 π$

Step 5.2

The resulting angle of $3.81672418$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 0.67513153 + 3.14159265$ .

$B = 3.81672418$

Step 6

Find the period of $\sin (B)$ .

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Step 6.1

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

$\frac{2 π}{| b |}$

Step 6.2

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

$\frac{2 π}{| 1 |}$

Step 6.3

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

$\frac{2 π}{1}$

Step 6.4

Divide $2 π$ by $1$ .

$2 π$

Step 7

Add $2 π$ to every negative angle to get positive angles.

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Step 7.1

Add $2 π$ to $- 0.67513153$ to find the positive angle.

$- 0.67513153 + 2 π$

Step 7.2

Subtract $0.67513153$ from $2 π$ .

$5.60805377$

Step 7.3

List the new angles.

$B = 5.60805377$

Step 8

The period of the $\sin (B)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$B = 3.81672418 + 2 π n, 5.60805377 + 2 π n$ , for any integer $n$