Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 12 \\ c = & 15 \\ a = & 9 \end{matrix} & \begin{matrix} A = \\ B = \\ C = & 90 \end{matrix} \end{matrix}$

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 $15$ of the triangle.

$B = \arcsin (\frac{12}{15})$

Step 1.3

Cancel the common factor of $12$ and $15$ .

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Step 1.3.1

Factor $3$ out of $12$ .

$B = \arcsin (\frac{3 (4)}{15})$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor $3$ out of $15$ .

$B = \arcsin (\frac{3 \cdot 4}{3 \cdot 5})$

Step 1.3.2.2

Cancel the common factor.

$B = \arcsin (\frac{3 \cdot 4}{3 \cdot 5})$

Step 1.3.2.3

Rewrite the expression.

$B = \arcsin (\frac{4}{5})$

Step 1.4

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

$B = 53.13010235$

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.

$A + 90 + 53.13010235 = 180$

Step 2.2

Solve the equation for $A$ .

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Step 2.2.1

Add $90$ and $53.13010235$ .

$A + 143.13010235 = 180$

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 $143.13010235$ from both sides of the equation.

$A = 180 - 143.13010235$

Step 2.2.2.2

Subtract $143.13010235$ from $180$ .

$A = 36.86989764$

Step 3

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

$A = 36.86989764$

$B = 53.13010235$

$C = 90$

$a = 9$

$b = 12$

$c = 15$