Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = & 10 \\ a = \end{matrix} & \begin{matrix} A = & 45 \\ B = & 45 \\ C = & 90 \end{matrix} \end{matrix}$

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.

$b = 10 \cdot \cos (45)$

Step 1.5

Cancel the common factor of $2$ .

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Step 1.5.1

Factor $2$ out of $10$ .

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

Step 1.5.2

Cancel the common factor.

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

Step 1.5.3

Rewrite the expression.

$b = 5 \cdot \sqrt{2}$

$b = 5 \sqrt{2}$

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.

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

Step 2.4

Simplify the expression.

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Step 2.4.1

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

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

Step 2.4.2

Apply the product rule to $5 \sqrt{2}$ .

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

Step 2.4.3

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

$a = \sqrt{100 - (25 {\sqrt{2}}^{2})}$

Step 2.5

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

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Step 2.5.1

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

$a = \sqrt{100 - (25 {(2^{\frac{1}{2}})}^{2})}$

Step 2.5.2

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

$a = \sqrt{100 - (25 \cdot 2^{\frac{1}{2} \cdot 2})}$

Step 2.5.3

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

$a = \sqrt{100 - (25 \cdot 2^{\frac{2}{2}})}$

Step 2.5.4

Cancel the common factor of $2$ .

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Step 2.5.4.1

Cancel the common factor.

$a = \sqrt{100 - (25 \cdot 2^{\frac{2}{2}})}$

Step 2.5.4.2

Rewrite the expression.

$a = \sqrt{100 - (25 \cdot 2)}$

Step 2.5.5

Evaluate the exponent.

$a = \sqrt{100 - (25 \cdot 2)}$

Step 2.6

Multiply $- (25 \cdot 2)$ .

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Step 2.6.1

Multiply $25$ by $2$ .

$a = \sqrt{100 - 1 \cdot 50}$

Step 2.6.2

Multiply $- 1$ by $50$ .

$a = \sqrt{100 - 50}$

Step 2.7

Subtract $50$ from $100$ .

$a = \sqrt{50}$

Step 2.8

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 2.8.1

Factor $25$ out of $50$ .

$a = \sqrt{25 (2)}$

Step 2.8.2

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

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

Step 2.9

Pull terms out from under the radical.

$a = 5 \sqrt{2}$

Step 3

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

$A = 45$

$B = 45$

$C = 90$

$a = 5 \sqrt{2}$

$b = 5 \sqrt{2}$

$c = 10$