Trigonometry Examples

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

Step 1

Use the law of cosines to find the unknown side of the triangle, given the other two sides and the included angle.

$c^{2} = a^{2} + b^{2} - 2 a b \cos (C)$

Step 2

Solve the equation.

$c = \sqrt{a^{2} + b^{2} - 2 a b \cos (C)}$

Step 3

Substitute the known values into the equation.

$c = \sqrt{{(5)}^{2} + {(12)}^{2} - 2 \cdot 5 \cdot 12 \cos (90 °)}$

Step 4

Simplify the results.

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Step 4.1

Raise $5$ to the power of $2$ .

$c = \sqrt{25 + {(12)}^{2} - 2 \cdot 5 \cdot (12 \cos (90 °))}$

Step 4.2

Raise $12$ to the power of $2$ .

$c = \sqrt{25 + 144 - 2 \cdot 5 \cdot (12 \cos (90 °))}$

Step 4.3

Multiply $- 2 \cdot 5 \cdot 12$ .

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Step 4.3.1

Multiply $- 2$ by $5$ .

$c = \sqrt{25 + 144 - 10 \cdot (12 \cos (90 °))}$

Step 4.3.2

Multiply $- 10$ by $12$ .

$c = \sqrt{25 + 144 - 120 \cos (90 °)}$

Step 4.4

The exact value of $\cos (90 °)$ is $0$ .

$c = \sqrt{25 + 144 - 120 \cdot 0}$

Step 4.5

Multiply $- 120$ by $0$ .

$c = \sqrt{25 + 144 + 0}$

Step 4.6

Add $25 + 144$ and $0$ .

$c = \sqrt{25 + 144}$

Step 4.7

Add $25$ and $144$ .

$c = \sqrt{169}$

Step 4.8

Rewrite $169$ as $13^{2}$ .

$c = \sqrt{13^{2}}$

Step 4.9

Pull terms out from under the radical, assuming positive real numbers.

$c = 13$

Step 5

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 6

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

$\frac{\sin (A)}{5} = \frac{\sin (90 °)}{13}$

Step 7

Solve the equation for $A$ .

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Step 7.1

Multiply both sides of the equation by $5$ .

$5 \frac{\sin (A)}{5} = 5 \frac{\sin (90 °)}{13}$

Step 7.2

Simplify both sides of the equation.

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Step 7.2.1

Simplify the left side.

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Step 7.2.1.1

Cancel the common factor of $5$ .

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Step 7.2.1.1.1

Cancel the common factor.

$5 \frac{\sin (A)}{5} = 5 \frac{\sin (90 °)}{13}$

Step 7.2.1.1.2

Rewrite the expression.

$\sin (A) = 5 \frac{\sin (90 °)}{13}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $5 \frac{\sin (90 °)}{13}$ .

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Step 7.2.2.1.1

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

$\sin (A) = 5 (\frac{1}{13})$

Step 7.2.2.1.2

Combine $5$ and $\frac{1}{13}$ .

$\sin (A) = \frac{5}{13}$

Step 7.3

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

$A = \arcsin (\frac{5}{13})$

Step 7.4

Simplify the right side.

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Step 7.4.1

Evaluate $\arcsin (\frac{5}{13})$ .

$A = 22.61986494$

Step 7.5

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.

$A = 180 - 22.61986494$

Step 7.6

Subtract $22.61986494$ from $180$ .

$A = 157.38013505$

Step 7.7

The solution to the equation $A = 22.61986494$ .

$A = 22.61986494, 157.38013505$

Step 7.8

Exclude the invalid angle.

$A = 22.61986494$

Step 8

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

$22.61986494 + 90 ° + B = 180$

Step 9

Solve the equation for $B$ .

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Step 9.1

Add $22.61986494$ and $90 °$ .

$112.61986494 + B = 180$

Step 9.2

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

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Step 9.2.1

Subtract $112.61986494$ from both sides of the equation.

$B = 180 - 112.61986494$

Step 9.2.2

Subtract $112.61986494$ from $180$ .

$B = 67.38013505$

Step 10

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

$A = 22.61986494$

$B = 67.38013505$

$C = 90 °$

$a = 5$

$b = 12$

$c = 13$