Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 9 \\ c = \\ a = & 4 \end{matrix} & \begin{matrix} A = \\ B = \\ C = & 90 \end{matrix} \end{matrix}$

Step 1

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

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 $c$ .

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

Step 1.3

Substitute the actual values into the equation.

$c = \sqrt{{(9)}^{2} + {(4)}^{2}}$

Step 1.4

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

$c = \sqrt{81 + {(4)}^{2}}$

Step 1.5

Raise $4$ to the power of $2$ .

$c = \sqrt{81 + 16}$

Step 1.6

Add $81$ and $16$ .

$c = \sqrt{97}$

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 $\sqrt{97}$ of the triangle.

$B = \arcsin (\frac{9}{\sqrt{97}})$

Step 2.3

Multiply $\frac{9}{\sqrt{97}}$ by $\frac{\sqrt{97}}{\sqrt{97}}$ .

$B = \arcsin (\frac{9}{\sqrt{97}} \cdot \frac{\sqrt{97}}{\sqrt{97}})$

Step 2.4

Combine and simplify the denominator.

Tap for more steps...

Step 2.4.1

Multiply $\frac{9}{\sqrt{97}}$ by $\frac{\sqrt{97}}{\sqrt{97}}$ .

$B = \arcsin (\frac{9 \sqrt{97}}{\sqrt{97} \sqrt{97}})$

Step 2.4.2

Raise $\sqrt{97}$ to the power of $1$ .

$B = \arcsin (\frac{9 \sqrt{97}}{\sqrt{97} \sqrt{97}})$

Step 2.4.3

Raise $\sqrt{97}$ to the power of $1$ .

$B = \arcsin (\frac{9 \sqrt{97}}{\sqrt{97} \sqrt{97}})$

Step 2.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$B = \arcsin (\frac{9 \sqrt{97}}{{\sqrt{97}}^{1 + 1}})$

Step 2.4.5

Add $1$ and $1$ .

$B = \arcsin (\frac{9 \sqrt{97}}{{\sqrt{97}}^{2}})$

Step 2.4.6

Rewrite ${\sqrt{97}}^{2}$ as $97$ .

Tap for more steps...

Step 2.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{97}$ as $97^{\frac{1}{2}}$ .

$B = \arcsin (\frac{9 \sqrt{97}}{{(97^{\frac{1}{2}})}^{2}})$

Step 2.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$B = \arcsin (\frac{9 \sqrt{97}}{97^{\frac{1}{2} \cdot 2}})$

Step 2.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$B = \arcsin (\frac{9 \sqrt{97}}{97^{\frac{2}{2}}})$

Step 2.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.4.6.4.1

Cancel the common factor.

$B = \arcsin (\frac{9 \sqrt{97}}{97^{\frac{2}{2}}})$

Step 2.4.6.4.2

Rewrite the expression.

$B = \arcsin (\frac{9 \sqrt{97}}{97})$

Step 2.4.6.5

Evaluate the exponent.

$B = \arcsin (\frac{9 \sqrt{97}}{97})$

Step 2.5

Evaluate $\arcsin (\frac{9 \sqrt{97}}{97})$ .

$B = 66.03751102$

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.

$A + 90 + 66.03751102 = 180$

Step 3.2

Solve the equation for $A$ .

Tap for more steps...

Step 3.2.1

Add $90$ and $66.03751102$ .

$A + 156.03751102 = 180$

Step 3.2.2

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

Tap for more steps...

Step 3.2.2.1

Subtract $156.03751102$ from both sides of the equation.

$A = 180 - 156.03751102$

Step 3.2.2.2

Subtract $156.03751102$ from $180$ .

$A = 23.96248897$

Step 4

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

$A = 23.96248897$

$B = 66.03751102$

$C = 90$

$a = 4$

$b = 9$

$c = \sqrt{97}$