Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = & 1 \\ a = \end{matrix} & \begin{matrix} A = & 30 \\ B = & 60 \\ C = & 90 \end{matrix} \end{matrix}$

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.

$b = 1 \cdot \cos (30)$

Step 1.5

Multiply $\frac{\sqrt{3}}{2}$ by $1$ .

$b = \frac{\sqrt{3}}{2}$

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.

$a = \sqrt{{(1)}^{2} - {(\frac{\sqrt{3}}{2})}^{2}}$

Step 2.4

Simplify the expression.

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Step 2.4.1

One to any power is one.

$a = \sqrt{1 - {(\frac{\sqrt{3}}{2})}^{2}}$

Step 2.4.2

Apply the product rule to $\frac{\sqrt{3}}{2}$ .

$a = \sqrt{1 - \frac{{\sqrt{3}}^{2}}{2^{2}}}$

Step 2.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$a = \sqrt{1 - \frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}}}$

Step 2.5.2

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

$a = \sqrt{1 - \frac{3^{\frac{1}{2} \cdot 2}}{2^{2}}}$

Step 2.5.3

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

$a = \sqrt{1 - \frac{3^{\frac{2}{2}}}{2^{2}}}$

Step 2.5.4

Cancel the common factor of $2$ .

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Step 2.5.4.1

Cancel the common factor.

$a = \sqrt{1 - \frac{3^{\frac{2}{2}}}{2^{2}}}$

Step 2.5.4.2

Rewrite the expression.

$a = \sqrt{1 - \frac{3}{2^{2}}}$

Step 2.5.5

Evaluate the exponent.

$a = \sqrt{1 - \frac{3}{2^{2}}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

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

$a = \sqrt{1 - \frac{3}{4}}$

Step 2.6.2

Write $1$ as a fraction with a common denominator.

$a = \sqrt{\frac{4}{4} - \frac{3}{4}}$

Step 2.6.3

Combine the numerators over the common denominator.

$a = \sqrt{\frac{4 - 3}{4}}$

Step 2.6.4

Subtract $3$ from $4$ .

$a = \sqrt{\frac{1}{4}}$

Step 2.7

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$a = \frac{\sqrt{1}}{\sqrt{4}}$

Step 2.8

Any root of $1$ is $1$ .

$a = \frac{1}{\sqrt{4}}$

Step 2.9

Simplify the denominator.

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Step 2.9.1

Rewrite $4$ as $2^{2}$ .

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

Step 2.9.2

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

$a = \frac{1}{2}$

Step 3

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

$A = 30$

$B = 60$

$C = 90$

$a = \frac{1}{2}$

$b = \frac{\sqrt{3}}{2}$

$c = 1$