Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 33.2 \\ c = \\ a = \end{matrix} & \begin{matrix} A = \\ B = & 61 \\ C = & 90 \end{matrix} \end{matrix}$

Step 1

Find the last angle of the triangle.

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Step 1.1

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

$A + 90 + 61 = 180$

Step 1.2

Solve the equation for $A$ .

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Step 1.2.1

Add $90$ and $61$ .

$A + 151 = 180$

Step 1.2.2

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

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Step 1.2.2.1

Subtract $151$ from both sides of the equation.

$A = 180 - 151$

Step 1.2.2.2

Subtract $151$ from $180$ .

$A = 29$

Step 2

Find $c$ .

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Step 2.1

The sine of an angle is equal to the ratio of the opposite side to the hypotenuse.

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

Step 2.2

Substitute the name of each side into the definition of the sine function.

$\sin (B) = \frac{b}{c}$

Step 2.3

Set up the equation to solve for the hypotenuse, in this case $c$ .

$c = \frac{b}{\sin (B)}$

Step 2.4

Substitute the values of each variable into the formula for sine.

$c = \frac{33.2}{\sin (61)}$

Step 2.5

Divide $33.2$ by $0.8746197$ .

$c = 37.95935505$

Step 3

Find the last side of the triangle using the Pythagorean theorem.

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Step 3.1

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 3.2

Solve the equation for $a$ .

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

Step 3.3

Substitute the actual values into the equation.

$a = \sqrt{{(37.95935505)}^{2} - {(33.2)}^{2}}$

Step 3.4

Raise $37.95935505$ to the power of $2$ .

$a = \sqrt{1440.91263606 - {(33.2)}^{2}}$

Step 3.5

Raise $33.2$ to the power of $2$ .

$a = \sqrt{1440.91263606 - 1 \cdot 1102.24}$

Step 3.6

Multiply $- 1$ by $1102.24$ .

$a = \sqrt{1440.91263606 - 1102.24}$

Step 3.7

Subtract $1102.24$ from $1440.91263606$ .

$a = \sqrt{338.67263606}$

Step 4

Convert $\sqrt{338.67263606}$ to a decimal.

$a = 18.4030605$

Step 5

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

$A = 29$

$B = 61$

$C = 90$

$a = 18.4030605$

$b = 33.2$

$c = 37.95935505$