Precalculus Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & \frac{51}{2} \\ c = & 17 \sqrt{3} \\ a = \end{matrix} & \begin{matrix} A = & 30 \\ B = & 60 \\ C = & 90 \end{matrix} \end{matrix}$

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 $a$ .

$a = \sqrt{c^{2} - b^{2}}$

Step 1.3

Substitute the actual values into the equation.

$a = \sqrt{{(17 \sqrt{3})}^{2} - {(\frac{51}{2})}^{2}}$

Step 1.4

Simplify the expression.

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Step 1.4.1

Apply the product rule to $17 \sqrt{3}$ .

$a = \sqrt{17^{2} {\sqrt{3}}^{2} - {(\frac{51}{2})}^{2}}$

Step 1.4.2

Raise $17$ to the power of $2$ .

$a = \sqrt{289 {\sqrt{3}}^{2} - {(\frac{51}{2})}^{2}}$

Step 1.5

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

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Step 1.5.1

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

$a = \sqrt{289 {(3^{\frac{1}{2}})}^{2} - {(\frac{51}{2})}^{2}}$

Step 1.5.2

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

$a = \sqrt{289 \cdot 3^{\frac{1}{2} \cdot 2} - {(\frac{51}{2})}^{2}}$

Step 1.5.3

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

$a = \sqrt{289 \cdot 3^{\frac{2}{2}} - {(\frac{51}{2})}^{2}}$

Step 1.5.4

Cancel the common factor of $2$ .

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Step 1.5.4.1

Cancel the common factor.

$a = \sqrt{289 \cdot 3^{\frac{2}{2}} - {(\frac{51}{2})}^{2}}$

Step 1.5.4.2

Rewrite the expression.

$a = \sqrt{289 \cdot 3 - {(\frac{51}{2})}^{2}}$

Step 1.5.5

Evaluate the exponent.

$a = \sqrt{289 \cdot 3 - {(\frac{51}{2})}^{2}}$

Step 1.6

Simplify the expression.

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Step 1.6.1

Multiply $289$ by $3$ .

$a = \sqrt{867 - {(\frac{51}{2})}^{2}}$

Step 1.6.2

Apply the product rule to $\frac{51}{2}$ .

$a = \sqrt{867 - \frac{51^{2}}{2^{2}}}$

Step 1.6.3

Raise $51$ to the power of $2$ .

$a = \sqrt{867 - \frac{2601}{2^{2}}}$

Step 1.6.4

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

$a = \sqrt{867 - \frac{2601}{4}}$

Step 1.7

To write $867$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$a = \sqrt{867 \cdot \frac{4}{4} - \frac{2601}{4}}$

Step 1.8

Combine $867$ and $\frac{4}{4}$ .

$a = \sqrt{\frac{867 \cdot 4}{4} - \frac{2601}{4}}$

Step 1.9

Combine the numerators over the common denominator.

$a = \sqrt{\frac{867 \cdot 4 - 2601}{4}}$

Step 1.10

Simplify the numerator.

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Step 1.10.1

Multiply $867$ by $4$ .

$a = \sqrt{\frac{3468 - 2601}{4}}$

Step 1.10.2

Subtract $2601$ from $3468$ .

$a = \sqrt{\frac{867}{4}}$

Step 1.11

Rewrite $\sqrt{\frac{867}{4}}$ as $\frac{\sqrt{867}}{\sqrt{4}}$ .

$a = \frac{\sqrt{867}}{\sqrt{4}}$

Step 1.12

Simplify the numerator.

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Step 1.12.1

Rewrite $867$ as $17^{2} \cdot 3$ .

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Step 1.12.1.1

Factor $289$ out of $867$ .

$a = \frac{\sqrt{289 (3)}}{\sqrt{4}}$

Step 1.12.1.2

Rewrite $289$ as $17^{2}$ .

$a = \frac{\sqrt{17^{2} \cdot 3}}{\sqrt{4}}$

Step 1.12.2

Pull terms out from under the radical.

$a = \frac{17 \sqrt{3}}{\sqrt{4}}$

Step 1.13

Simplify the denominator.

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Step 1.13.1

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

$a = \frac{17 \sqrt{3}}{\sqrt{2^{2}}}$

Step 1.13.2

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

$a = \frac{17 \sqrt{3}}{2}$

Step 2

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

$A = 30$

$B = 60$

$C = 90$

$a = \frac{17 \sqrt{3}}{2}$

$b = \frac{51}{2}$

$c = 17 \sqrt{3}$