Trigonometry Examples

$A = 90$ , $a = 18$ , $b = 9$

Step 1

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

Tap for more steps...

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 $c$ .

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

Step 1.3

Substitute the actual values into the equation.

$c = \sqrt{{(18)}^{2} - {(9)}^{2}}$

Step 1.4

Raise $18$ to the power of $2$ .

$c = \sqrt{324 - {(9)}^{2}}$

Step 1.5

Raise $9$ to the power of $2$ .

$c = \sqrt{324 - 1 \cdot 81}$

Step 1.6

Multiply $- 1$ by $81$ .

$c = \sqrt{324 - 81}$

Step 1.7

Subtract $81$ from $324$ .

$c = \sqrt{243}$

Step 1.8

Rewrite $243$ as $9^{2} \cdot 3$ .

Tap for more steps...

Step 1.8.1

Factor $81$ out of $243$ .

$c = \sqrt{81 (3)}$

Step 1.8.2

Rewrite $81$ as $9^{2}$ .

$c = \sqrt{9^{2} \cdot 3}$

Step 1.9

Pull terms out from under the radical.

$c = 9 \sqrt{3}$

Step 2

Find $B$ .

Tap for more steps...

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 $18$ of the triangle.

$B = \arcsin (\frac{9}{18})$

Step 2.3

Cancel the common factor of $9$ and $18$ .

Tap for more steps...

Step 2.3.1

Factor $9$ out of $9$ .

$B = \arcsin (\frac{9 (1)}{18})$

Step 2.3.2

Cancel the common factors.

Tap for more steps...

Step 2.3.2.1

Factor $9$ out of $18$ .

$B = \arcsin (\frac{9 \cdot 1}{9 \cdot 2})$

Step 2.3.2.2

Cancel the common factor.

$B = \arcsin (\frac{9 \cdot 1}{9 \cdot 2})$

Step 2.3.2.3

Rewrite the expression.

$B = \arcsin (\frac{1}{2})$

Step 2.4

The exact value of $\arcsin (\frac{1}{2})$ is $30$ .

$B = 30$

Step 3

Find the last angle of the triangle.

Tap for more steps...

Step 3.1

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

$90 + C + 30 = 180$

Step 3.2

Solve the equation for $C$ .

Tap for more steps...

Step 3.2.1

Add $90$ and $30$ .

$C + 120 = 180$

Step 3.2.2

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

Tap for more steps...

Step 3.2.2.1

Subtract $120$ from both sides of the equation.

$C = 180 - 120$

Step 3.2.2.2

Subtract $120$ from $180$ .

$C = 60$

Step 4

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

$A = 90$

$B = 30$

$C = 60$

$a = 18$

$b = 9$

$c = 9 \sqrt{3}$