Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 9 \\ c = \\ a = & 6 \end{matrix} & \begin{matrix} A = \\ B = \\ C = & 90 \end{matrix} \end{matrix}$

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.

$c = \sqrt{{(9)}^{2} + {(6)}^{2}}$

Step 1.4

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

$c = \sqrt{81 + {(6)}^{2}}$

Step 1.5

Raise $6$ to the power of $2$ .

$c = \sqrt{81 + 36}$

Step 1.6

Add $81$ and $36$ .

$c = \sqrt{117}$

Step 1.7

Rewrite $117$ as $3^{2} \cdot 13$ .

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Step 1.7.1

Factor $9$ out of $117$ .

$c = \sqrt{9 (13)}$

Step 1.7.2

Rewrite $9$ as $3^{2}$ .

$c = \sqrt{3^{2} \cdot 13}$

Step 1.8

Pull terms out from under the radical.

$c = 3 \sqrt{13}$

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 $3 \sqrt{13}$ of the triangle.

$B = \arcsin (\frac{9}{3 \sqrt{13}})$

Step 2.3

Cancel the common factor of $9$ and $3$ .

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Step 2.3.1

Factor $3$ out of $9$ .

$B = \arcsin (\frac{3 \cdot 3}{3 \sqrt{13}})$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $3$ out of $3 \sqrt{13}$ .

$B = \arcsin (\frac{3 \cdot 3}{3 (\sqrt{13})})$

Step 2.3.2.2

Cancel the common factor.

$B = \arcsin (\frac{3 \cdot 3}{3 \sqrt{13}})$

Step 2.3.2.3

Rewrite the expression.

$B = \arcsin (\frac{3}{\sqrt{13}})$

Step 2.4

Multiply $\frac{3}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$B = \arcsin (\frac{3}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}})$

Step 2.5

Combine and simplify the denominator.

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Step 2.5.1

Multiply $\frac{3}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$B = \arcsin (\frac{3 \sqrt{13}}{\sqrt{13} \sqrt{13}})$

Step 2.5.2

Raise $\sqrt{13}$ to the power of $1$ .

$B = \arcsin (\frac{3 \sqrt{13}}{\sqrt{13} \sqrt{13}})$

Step 2.5.3

Raise $\sqrt{13}$ to the power of $1$ .

$B = \arcsin (\frac{3 \sqrt{13}}{\sqrt{13} \sqrt{13}})$

Step 2.5.4

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

$B = \arcsin (\frac{3 \sqrt{13}}{{\sqrt{13}}^{1 + 1}})$

Step 2.5.5

Add $1$ and $1$ .

$B = \arcsin (\frac{3 \sqrt{13}}{{\sqrt{13}}^{2}})$

Step 2.5.6

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 2.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$B = \arcsin (\frac{3 \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}})$

Step 2.5.6.2

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

$B = \arcsin (\frac{3 \sqrt{13}}{13^{\frac{1}{2} \cdot 2}})$

Step 2.5.6.3

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

$B = \arcsin (\frac{3 \sqrt{13}}{13^{\frac{2}{2}}})$

Step 2.5.6.4

Cancel the common factor of $2$ .

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Step 2.5.6.4.1

Cancel the common factor.

$B = \arcsin (\frac{3 \sqrt{13}}{13^{\frac{2}{2}}})$

Step 2.5.6.4.2

Rewrite the expression.

$B = \arcsin (\frac{3 \sqrt{13}}{13})$

Step 2.5.6.5

Evaluate the exponent.

$B = \arcsin (\frac{3 \sqrt{13}}{13})$

Step 2.6

Evaluate $\arcsin (\frac{3 \sqrt{13}}{13})$ .

$B = 56.30993247$

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.

$A + 90 + 56.30993247 = 180$

Step 3.2

Solve the equation for $A$ .

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Step 3.2.1

Add $90$ and $56.30993247$ .

$A + 146.30993247 = 180$

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 $146.30993247$ from both sides of the equation.

$A = 180 - 146.30993247$

Step 3.2.2.2

Subtract $146.30993247$ from $180$ .

$A = 33.69006752$

Step 4

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

$A = 33.69006752$

$B = 56.30993247$

$C = 90$

$a = 6$

$b = 9$

$c = 3 \sqrt{13}$