Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = & 10 \\ a = & 6 \end{matrix} & \begin{matrix} A = \\ B = \\ C = \end{matrix} \end{matrix}$

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 $b$ .

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

Step 2.3

Substitute the actual values into the equation.

$b = \sqrt{{(10)}^{2} - {(6)}^{2}}$

Step 2.4

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

$b = \sqrt{100 - {(6)}^{2}}$

Step 2.5

Raise $6$ to the power of $2$ .

$b = \sqrt{100 - 1 \cdot 36}$

Step 2.6

Multiply $- 1$ by $36$ .

$b = \sqrt{100 - 36}$

Step 2.7

Subtract $36$ from $100$ .

$b = \sqrt{64}$

Step 2.8

Rewrite $64$ as $8^{2}$ .

$b = \sqrt{8^{2}}$

Step 2.9

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

$b = 8$

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 $10$ of the triangle.

$B = \arcsin (\frac{8}{10})$

Step 3.3

Cancel the common factor of $8$ and $10$ .

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Step 3.3.1

Factor $2$ out of $8$ .

$B = \arcsin (\frac{2 (4)}{10})$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Factor $2$ out of $10$ .

$B = \arcsin (\frac{2 \cdot 4}{2 \cdot 5})$

Step 3.3.2.2

Cancel the common factor.

$B = \arcsin (\frac{2 \cdot 4}{2 \cdot 5})$

Step 3.3.2.3

Rewrite the expression.

$B = \arcsin (\frac{4}{5})$

Step 3.4

Evaluate $\arcsin (\frac{4}{5})$ .

$B = 53.13010235$

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.

$A + 90 + 53.13010235 = 180$

Step 4.2

Solve the equation for $A$ .

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Step 4.2.1

Add $90$ and $53.13010235$ .

$A + 143.13010235 = 180$

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 $143.13010235$ from both sides of the equation.

$A = 180 - 143.13010235$

Step 4.2.2.2

Subtract $143.13010235$ from $180$ .

$A = 36.86989764$

Step 5

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

$A = 36.86989764$

$B = 53.13010235$

$C = 90$

$a = 6$

$b = 8$

$c = 10$