Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = & 27 \\ a = \end{matrix} & \begin{matrix} A = \\ B = & 54 \\ C = & 90 \end{matrix} \end{matrix}$

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.

$A + 90 + 54 = 180$

Step 1.2

Solve the equation for $A$ .

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Step 1.2.1

Add $90$ and $54$ .

$A + 144 = 180$

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 $144$ from both sides of the equation.

$A = 180 - 144$

Step 1.2.2.2

Subtract $144$ from $180$ .

$A = 36$

Step 2

Find $b$ .

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Step 2.1

The cosine of an angle is equal to the ratio of the adjacent side to the hypotenuse.

$\cos (A) = \frac{adj}{hyp}$

Step 2.2

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

$\cos (A) = \frac{b}{c}$

Step 2.3

Set up the equation to solve for the adjacent side, in this case $b$ .

$b = c \cdot \cos (A)$

Step 2.4

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

$b = 27 \cdot \cos (36)$

Step 2.5

Multiply $27$ by $0.80901699$ .

$b = 21.84345884$

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.

$a = \sqrt{{(27)}^{2} - {(21.84345884)}^{2}}$

Step 3.4

Raise $27$ to the power of $2$ .

$a = \sqrt{729 - {(21.84345884)}^{2}}$

Step 3.5

Raise $21.84345884$ to the power of $2$ .

$a = \sqrt{729 - 1 \cdot 477.13669444}$

Step 3.6

Multiply $- 1$ by $477.13669444$ .

$a = \sqrt{729 - 477.13669444}$

Step 3.7

Subtract $477.13669444$ from $729$ .

$a = \sqrt{251.86330555}$

Step 4

Convert $\sqrt{251.86330555}$ to a decimal.

$a = 15.87020181$

Step 5

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

$A = 36$

$B = 54$

$C = 90$

$a = 15.87020181$

$b = 21.84345884$

$c = 27$