Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 10 \\ c = \\ a = & 5 \end{matrix} & \begin{matrix} A = \\ B = \\ C = \end{matrix} \end{matrix}$

Step 1

Assume that angle $C = 90$ .

$C = 90$

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 $c$ .

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

Step 2.3

Substitute the actual values into the equation.

$c = \sqrt{{(10)}^{2} + {(5)}^{2}}$

Step 2.4

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

$c = \sqrt{100 + {(5)}^{2}}$

Step 2.5

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

$c = \sqrt{100 + 25}$

Step 2.6

Add $100$ and $25$ .

$c = \sqrt{125}$

Step 2.7

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 2.7.1

Factor $25$ out of $125$ .

$c = \sqrt{25 (5)}$

Step 2.7.2

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

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

Step 2.8

Pull terms out from under the radical.

$c = 5 \sqrt{5}$

Step 3

Find $B$ .

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Step 3.1

The angle $B$ can be found using the inverse sine function.

$B = \arcsin (\frac{opp}{hyp})$

Step 3.2

Substitute in the values of the opposite side to angle $B$ and hypotenuse $5 \sqrt{5}$ of the triangle.

$B = \arcsin (\frac{10}{5 \sqrt{5}})$

Step 3.3

Cancel the common factor of $10$ and $5$ .

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Step 3.3.1

Factor $5$ out of $10$ .

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

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Factor $5$ out of $5 \sqrt{5}$ .

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

Step 3.3.2.2

Cancel the common factor.

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

Step 3.3.2.3

Rewrite the expression.

$B = \arcsin (\frac{2}{\sqrt{5}})$

Step 3.4

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Step 3.5

Combine and simplify the denominator.

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Step 3.5.1

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$B = \arcsin (\frac{2 \sqrt{5}}{\sqrt{5} \sqrt{5}})$

Step 3.5.2

Raise $\sqrt{5}$ to the power of $1$ .

$B = \arcsin (\frac{2 \sqrt{5}}{\sqrt{5} \sqrt{5}})$

Step 3.5.3

Raise $\sqrt{5}$ to the power of $1$ .

$B = \arcsin (\frac{2 \sqrt{5}}{\sqrt{5} \sqrt{5}})$

Step 3.5.4

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

$B = \arcsin (\frac{2 \sqrt{5}}{{\sqrt{5}}^{1 + 1}})$

Step 3.5.5

Add $1$ and $1$ .

$B = \arcsin (\frac{2 \sqrt{5}}{{\sqrt{5}}^{2}})$

Step 3.5.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$B = \arcsin (\frac{2 \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}})$

Step 3.5.6.2

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

$B = \arcsin (\frac{2 \sqrt{5}}{5^{\frac{1}{2} \cdot 2}})$

Step 3.5.6.3

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

$B = \arcsin (\frac{2 \sqrt{5}}{5^{\frac{2}{2}}})$

Step 3.5.6.4

Cancel the common factor of $2$ .

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Step 3.5.6.4.1

Cancel the common factor.

$B = \arcsin (\frac{2 \sqrt{5}}{5^{\frac{2}{2}}})$

Step 3.5.6.4.2

Rewrite the expression.

$B = \arcsin (\frac{2 \sqrt{5}}{5})$

Step 3.5.6.5

Evaluate the exponent.

$B = \arcsin (\frac{2 \sqrt{5}}{5})$

Step 3.6

Evaluate $\arcsin (\frac{2 \sqrt{5}}{5})$ .

$B = 63.43494882$

Step 4

Find the last angle of the triangle.

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Step 4.1

The sum of all the angles in a triangle is $180$ degrees.

$A + 90 + 63.43494882 = 180$

Step 4.2

Solve the equation for $A$ .

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Step 4.2.1

Add $90$ and $63.43494882$ .

$A + 153.43494882 = 180$

Step 4.2.2

Move all terms not containing $A$ to the right side of the equation.

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Step 4.2.2.1

Subtract $153.43494882$ from both sides of the equation.

$A = 180 - 153.43494882$

Step 4.2.2.2

Subtract $153.43494882$ from $180$ .

$A = 26.56505117$

Step 5

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

$A = 26.56505117$

$B = 63.43494882$

$C = 90$

$a = 5$

$b = 10$

$c = 5 \sqrt{5}$