Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 36 \\ c = & 39 \\ a = & 15 \end{matrix} & \begin{matrix} A = \\ B = \\ C = & 90 \end{matrix} \end{matrix}$

Step 1

Find $B$ .

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Step 1.1

The angle $B$ can be found using the inverse sine function.

$B = \arcsin (\frac{opp}{hyp})$

Step 1.2

Substitute in the values of the opposite side to angle $B$ and hypotenuse $39$ of the triangle.

$B = \arcsin (\frac{36}{39})$

Step 1.3

Cancel the common factor of $36$ and $39$ .

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Step 1.3.1

Factor $3$ out of $36$ .

$B = \arcsin (\frac{3 (12)}{39})$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor $3$ out of $39$ .

$B = \arcsin (\frac{3 \cdot 12}{3 \cdot 13})$

Step 1.3.2.2

Cancel the common factor.

$B = \arcsin (\frac{3 \cdot 12}{3 \cdot 13})$

Step 1.3.2.3

Rewrite the expression.

$B = \arcsin (\frac{12}{13})$

Step 1.4

Evaluate $\arcsin (\frac{12}{13})$ .

$B = 67.38013505$

Step 2

Find the last angle of the triangle.

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Step 2.1

The sum of all the angles in a triangle is $180$ degrees.

$A + 90 + 67.38013505 = 180$

Step 2.2

Solve the equation for $A$ .

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Step 2.2.1

Add $90$ and $67.38013505$ .

$A + 157.38013505 = 180$

Step 2.2.2

Move all terms not containing $A$ to the right side of the equation.

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Step 2.2.2.1

Subtract $157.38013505$ from both sides of the equation.

$A = 180 - 157.38013505$

Step 2.2.2.2

Subtract $157.38013505$ from $180$ .

$A = 22.61986494$

Step 3

These are the results for all angles and sides for the given triangle.

$A = 22.61986494$

$B = 67.38013505$

$C = 90$

$a = 15$

$b = 36$

$c = 39$