Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 15 \\ c = \\ a = & 20 \end{matrix} & \begin{matrix} A = \\ B = \\ C = & 90 \end{matrix} \end{matrix}$

Step 1

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

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Step 1.1

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 1.2

Solve the equation for $c$ .

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

Step 1.3

Substitute the actual values into the equation.

$c = \sqrt{{(15)}^{2} + {(20)}^{2}}$

Step 1.4

Raise $15$ to the power of $2$ .

$c = \sqrt{225 + {(20)}^{2}}$

Step 1.5

Raise $20$ to the power of $2$ .

$c = \sqrt{225 + 400}$

Step 1.6

Add $225$ and $400$ .

$c = \sqrt{625}$

Step 1.7

Rewrite $625$ as $25^{2}$ .

$c = \sqrt{25^{2}}$

Step 1.8

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

$c = 25$

Step 2

Find $B$ .

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Step 2.1

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

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

Step 2.2

Substitute in the values of the opposite side to angle $B$ and hypotenuse $25$ of the triangle.

$B = \arcsin (\frac{15}{25})$

Step 2.3

Cancel the common factor of $15$ and $25$ .

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Step 2.3.1

Factor $5$ out of $15$ .

$B = \arcsin (\frac{5 (3)}{25})$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $5$ out of $25$ .

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

Step 2.3.2.2

Cancel the common factor.

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

Step 2.3.2.3

Rewrite the expression.

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

Step 2.4

Evaluate $\arcsin (\frac{3}{5})$ .

$B = 36.86989764$

Step 3

Find the last angle of the triangle.

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Step 3.1

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

$A + 90 + 36.86989764 = 180$

Step 3.2

Solve the equation for $A$ .

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Step 3.2.1

Add $90$ and $36.86989764$ .

$A + 126.86989764 = 180$

Step 3.2.2

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

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Step 3.2.2.1

Subtract $126.86989764$ from both sides of the equation.

$A = 180 - 126.86989764$

Step 3.2.2.2

Subtract $126.86989764$ from $180$ .

$A = 53.13010235$

Step 4

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

$A = 53.13010235$

$B = 36.86989764$

$C = 90$

$a = 20$

$b = 15$

$c = 25$