Trigonometry Examples

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$\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = & 9 \\ a = & 7 \end{matrix} & \begin{matrix} A = \\ B = \\ C = & 90 \end{matrix} \end{matrix}$

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

$b$ .

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

Step 1.3

Substitute the actual values into the equation.

$b = \sqrt{{(9)}^{2} - {(7)}^{2}}$

Step 1.4

Raise

$9$ to the power of

$2$ .

$b = \sqrt{81 - {(7)}^{2}}$

Step 1.5

Raise

$7$ to the power of

$2$ .

$b = \sqrt{81 - 1 \cdot 49}$

Step 1.6

Multiply

$- 1$ by

$49$ .

$b = \sqrt{81 - 49}$

Step 1.7

Subtract

$49$ from

$81$ .

$b = \sqrt{32}$

Step 1.8

Rewrite

$32$ as

$4^{2} \cdot 2$ .

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Step 1.8.1

Factor

$16$ out of

$32$ .

$b = \sqrt{16 (2)}$

Step 1.8.2

Rewrite

$16$ as

$4^{2}$ .

$b = \sqrt{4^{2} \cdot 2}$

Step 1.9

Pull terms out from under the radical.

$b = 4 \sqrt{2}$

Step 2

Find

$B$ .

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Step 2.1

The angle

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

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

Step 2.2

Substitute in the values of the opposite side to angle

$B$ and hypotenuse

$9$ of the triangle.

$B = \arcsin (\frac{4 \sqrt{2}}{9})$

Step 2.3

Evaluate

$\arcsin (\frac{4 \sqrt{2}}{9})$ .

$B = 38.94244126$

Step 3

Find the last angle of the triangle.

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Step 3.1

The sum of all the angles in a triangle is

$180$ degrees.

$A + 90 + 38.94244126 = 180$

Step 3.2

Solve the equation for

$A$ .

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Step 3.2.1

Add

$90$ and

$38.94244126$ .

$A + 128.94244126 = 180$

Step 3.2.2

Move all terms not containing

$A$ to the right side of the equation.

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Step 3.2.2.1

Subtract

$128.94244126$ from both sides of the equation.

$A = 180 - 128.94244126$

Step 3.2.2.2

Subtract

$128.94244126$ from

$180$ .

$A = 51.05755873$

Step 4

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

$A = 51.05755873$

$B = 38.94244126$

$C = 90$

$a = 7$

$b = 4 \sqrt{2}$

$c = 9$