Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = & 9 \\ a = \end{matrix} & \begin{matrix} A = & 45 \\ B = & 45 \\ C = & 90 \end{matrix} \end{matrix}$

Step 1

Find $b$ .

Tap for more steps...

Step 1.1

The cosine of an angle is equal to the ratio of the adjacent side to the hypotenuse.

$\cos (A) = \frac{adj}{hyp}$

Step 1.2

Substitute the name of each side into the definition of the cosine function.

$\cos (A) = \frac{b}{c}$

Step 1.3

Set up the equation to solve for the adjacent side, in this case $b$ .

$b = c \cdot \cos (A)$

Step 1.4

Substitute the values of each variable into the formula for cosine.

$b = 9 \cdot \cos (45)$

Step 1.5

Combine $9$ and $\frac{\sqrt{2}}{2}$ .

$b = \frac{9 \sqrt{2}}{2}$

Step 2

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

Tap for more steps...

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.

$a = \sqrt{{(9)}^{2} - {(\frac{9 \sqrt{2}}{2})}^{2}}$

Step 2.4

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

$a = \sqrt{81 - {(\frac{9 \sqrt{2}}{2})}^{2}}$

Step 2.5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 2.5.1

Apply the product rule to $\frac{9 \sqrt{2}}{2}$ .

$a = \sqrt{81 - \frac{{(9 \sqrt{2})}^{2}}{2^{2}}}$

Step 2.5.2

Apply the product rule to $9 \sqrt{2}$ .

$a = \sqrt{81 - \frac{9^{2} {\sqrt{2}}^{2}}{2^{2}}}$

Step 2.6

Simplify the numerator.

Tap for more steps...

Step 2.6.1

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

$a = \sqrt{81 - \frac{81 {\sqrt{2}}^{2}}{2^{2}}}$

Step 2.6.2

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

Tap for more steps...

Step 2.6.2.1

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

$a = \sqrt{81 - \frac{81 {(2^{\frac{1}{2}})}^{2}}{2^{2}}}$

Step 2.6.2.2

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

$a = \sqrt{81 - \frac{81 \cdot 2^{\frac{1}{2} \cdot 2}}{2^{2}}}$

Step 2.6.2.3

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

$a = \sqrt{81 - \frac{81 \cdot 2^{\frac{2}{2}}}{2^{2}}}$

Step 2.6.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.6.2.4.1

Cancel the common factor.

$a = \sqrt{81 - \frac{81 \cdot 2^{\frac{2}{2}}}{2^{2}}}$

Step 2.6.2.4.2

Rewrite the expression.

$a = \sqrt{81 - \frac{81 \cdot 2}{2^{2}}}$

Step 2.6.2.5

Evaluate the exponent.

$a = \sqrt{81 - \frac{81 \cdot 2}{2^{2}}}$

Step 2.7

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 2.7.1

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

$a = \sqrt{81 - \frac{81 \cdot 2}{4}}$

Step 2.7.2

Multiply $81$ by $2$ .

$a = \sqrt{81 - \frac{162}{4}}$

Step 2.7.3

Cancel the common factor of $162$ and $4$ .

Tap for more steps...

Step 2.7.3.1

Factor $2$ out of $162$ .

$a = \sqrt{81 - \frac{2 (81)}{4}}$

Step 2.7.3.2

Cancel the common factors.

Tap for more steps...

Step 2.7.3.2.1

Factor $2$ out of $4$ .

$a = \sqrt{81 - \frac{2 \cdot 81}{2 \cdot 2}}$

Step 2.7.3.2.2

Cancel the common factor.

$a = \sqrt{81 - \frac{2 \cdot 81}{2 \cdot 2}}$

Step 2.7.3.2.3

Rewrite the expression.

$a = \sqrt{81 - \frac{81}{2}}$

Step 2.8

To write $81$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$a = \sqrt{81 \cdot \frac{2}{2} - \frac{81}{2}}$

Step 2.9

Combine $81$ and $\frac{2}{2}$ .

$a = \sqrt{\frac{81 \cdot 2}{2} - \frac{81}{2}}$

Step 2.10

Combine the numerators over the common denominator.

$a = \sqrt{\frac{81 \cdot 2 - 81}{2}}$

Step 2.11

Simplify the numerator.

Tap for more steps...

Step 2.11.1

Multiply $81$ by $2$ .

$a = \sqrt{\frac{162 - 81}{2}}$

Step 2.11.2

Subtract $81$ from $162$ .

$a = \sqrt{\frac{81}{2}}$

Step 2.12

Rewrite $\sqrt{\frac{81}{2}}$ as $\frac{\sqrt{81}}{\sqrt{2}}$ .

$a = \frac{\sqrt{81}}{\sqrt{2}}$

Step 2.13

Simplify the numerator.

Tap for more steps...

Step 2.13.1

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

$a = \frac{\sqrt{9^{2}}}{\sqrt{2}}$

Step 2.13.2

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

$a = \frac{9}{\sqrt{2}}$

Step 2.14

Multiply $\frac{9}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$a = \frac{9}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 2.15

Combine and simplify the denominator.

Tap for more steps...

Step 2.15.1

Multiply $\frac{9}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$a = \frac{9 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 2.15.2

Raise $\sqrt{2}$ to the power of $1$ .

$a = \frac{9 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 2.15.3

Raise $\sqrt{2}$ to the power of $1$ .

$a = \frac{9 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 2.15.4

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

$a = \frac{9 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 2.15.5

Add $1$ and $1$ .

$a = \frac{9 \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 2.15.6

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

Tap for more steps...

Step 2.15.6.1

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

$a = \frac{9 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 2.15.6.2

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

$a = \frac{9 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 2.15.6.3

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

$a = \frac{9 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.15.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.15.6.4.1

Cancel the common factor.

$a = \frac{9 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.15.6.4.2

Rewrite the expression.

$a = \frac{9 \sqrt{2}}{2}$

Step 2.15.6.5

Evaluate the exponent.

$a = \frac{9 \sqrt{2}}{2}$

Step 3

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

$A = 45$

$B = 45$

$C = 90$

$a = \frac{9 \sqrt{2}}{2}$

$b = \frac{9 \sqrt{2}}{2}$

$c = 9$