Trigonometry Examples

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

Step 1

Assume that angle

$C = 90$ .

$C = 90$

Step 2

Find the last side of the triangle using the Pythagorean theorem.

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Step 2.1

Use the Pythagorean theorem to find the unknown side. In any right triangle, the area of the square whose side is the hypotenuse (the side of a right triangle opposite the right angle) is equal to the sum of areas of the squares whose sides are the two legs (the two sides other than the hypotenuse).

$a^{2} + b^{2} = c^{2}$

Step 2.2

Solve the equation for

$a$ .

$a = \sqrt{c^{2} - b^{2}}$

Step 2.3

Substitute the actual values into the equation.

$a = \sqrt{{(13)}^{2} - {(5)}^{2}}$

Step 2.4

Raise

$13$ to the power of

$2$ .

$a = \sqrt{169 - {(5)}^{2}}$

Step 2.5

Raise

$5$ to the power of

$2$ .

$a = \sqrt{169 - 1 \cdot 25}$

Step 2.6

Multiply

$- 1$ by

$25$ .

$a = \sqrt{169 - 25}$

Step 2.7

Subtract

$25$ from

$169$ .

$a = \sqrt{144}$

Step 2.8

Rewrite

$144$ as

$12^{2}$ .

$a = \sqrt{12^{2}}$

Step 2.9

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

$a = 12$

Step 3

Find

$B$ .

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Step 3.1

The angle

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

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

Step 3.2

Substitute in the values of the opposite side to angle

$B$ and hypotenuse

$13$ of the triangle.

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

Step 3.3

Evaluate

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

$B = 22.61986494$

Step 4

Find the last angle of the triangle.

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Step 4.1

The sum of all the angles in a triangle is

$180$ degrees.

$A + 90 + 22.61986494 = 180$

Step 4.2

Solve the equation for

$A$ .

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Step 4.2.1

Add

$90$ and

$22.61986494$ .

$A + 112.61986494 = 180$

Step 4.2.2

Move all terms not containing

$A$ to the right side of the equation.

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Step 4.2.2.1

Subtract

$112.61986494$ from both sides of the equation.

$A = 180 - 112.61986494$

Step 4.2.2.2

Subtract

$112.61986494$ from

$180$ .

$A = 67.38013505$

Step 5

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

$A = 67.38013505$

$B = 22.61986494$

$C = 90$

$a = 12$

$b = 5$

$c = 13$