Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 5 \\ c = & 13 \\ a = & 12 \end{matrix} & \begin{matrix} A = \\ B = \\ C = & 90 ° \end{matrix} \end{matrix}$

Step 1

The law of sines is based on the proportionality of sides and angles in triangles. The law states that for the angles of a non-right triangle, each angle of the triangle has the same ratio of angle measure to sine value.

$\frac{\sin (A)}{a} = \frac{\sin (B)}{b} = \frac{\sin (C)}{c}$

Step 2

Substitute the known values into the law of sines to find $A$ .

$\frac{\sin (A)}{12} = \frac{\sin (90 °)}{13}$

Step 3

Solve the equation for $A$ .

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Step 3.1

Multiply both sides of the equation by $12$ .

$12 \frac{\sin (A)}{12} = 12 \frac{\sin (90 °)}{13}$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Cancel the common factor of $12$ .

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Step 3.2.1.1.1

Cancel the common factor.

$12 \frac{\sin (A)}{12} = 12 \frac{\sin (90 °)}{13}$

Step 3.2.1.1.2

Rewrite the expression.

$\sin (A) = 12 \frac{\sin (90 °)}{13}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $12 \frac{\sin (90 °)}{13}$ .

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Step 3.2.2.1.1

The exact value of $\sin (90 °)$ is $1$ .

$\sin (A) = 12 (\frac{1}{13})$

Step 3.2.2.1.2

Combine $12$ and $\frac{1}{13}$ .

$\sin (A) = \frac{12}{13}$

Step 3.3

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

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

Step 3.4

Simplify the right side.

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Step 3.4.1

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

$A = 67.38013505$

Step 3.5

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.

$A = 180 - 67.38013505$

Step 3.6

Subtract $67.38013505$ from $180$ .

$A = 112.61986494$

Step 3.7

The solution to the equation $A = 67.38013505$ .

$A = 67.38013505, 112.61986494$

Step 3.8

Exclude the invalid angle.

$A = 67.38013505$

Step 4

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

$67.38013505 + 90 ° + B = 180$

Step 5

Solve the equation for $B$ .

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Step 5.1

Add $67.38013505$ and $90 °$ .

$157.38013505 + B = 180$

Step 5.2

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

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Step 5.2.1

Subtract $157.38013505$ from both sides of the equation.

$B = 180 - 157.38013505$

Step 5.2.2

Subtract $157.38013505$ from $180$ .

$B = 22.61986494$

Step 6

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

$A = 67.38013505$

$B = 22.61986494$

$C = 90 °$

$a = 12$

$b = 5$

$c = 13$