Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = & 10 \\ a = \end{matrix} & \begin{matrix} A = & 60 \\ B = & 30 \\ C = & 90 \end{matrix} \end{matrix}$

Step 1

Find $b$ .

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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.

$b = 10 \cdot \cos (60)$

Step 1.5

Cancel the common factor of $2$ .

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Step 1.5.1

Factor $2$ out of $10$ .

$b = 2 (5) \cdot \frac{1}{2}$

Step 1.5.2

Cancel the common factor.

$b = 2 \cdot 5 \cdot \frac{1}{2}$

Step 1.5.3

Rewrite the expression.

$b = 5$

Step 2

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

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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.

$a = \sqrt{{(10)}^{2} - {(5)}^{2}}$

Step 2.4

Raise $10$ to the power of $2$ .

$a = \sqrt{100 - {(5)}^{2}}$

Step 2.5

Raise $5$ to the power of $2$ .

$a = \sqrt{100 - 1 \cdot 25}$

Step 2.6

Multiply $- 1$ by $25$ .

$a = \sqrt{100 - 25}$

Step 2.7

Subtract $25$ from $100$ .

$a = \sqrt{75}$

Step 2.8

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 2.8.1

Factor $25$ out of $75$ .

$a = \sqrt{25 (3)}$

Step 2.8.2

Rewrite $25$ as $5^{2}$ .

$a = \sqrt{5^{2} \cdot 3}$

Step 2.9

Pull terms out from under the radical.

$a = 5 \sqrt{3}$

Step 3

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

$A = 60$

$B = 30$

$C = 90$

$a = 5 \sqrt{3}$

$b = 5$

$c = 10$