Trigonometry Examples

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

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

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

$180$ degrees.

A + 90 + 54 = 180

$A + 90 + 54 = 180$

Step 1.2

Solve the equation for

A

$A$ .

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Step 1.2.1

Add

90

$90$ and

54

$54$ .

A + 144 = 180

$A + 144 = 180$

Step 1.2.2

Move all terms not containing

A

$A$ to the right side of the equation.

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Step 1.2.2.1

Subtract

144

$144$ from both sides of the equation.

A = 180 - 144

$A = 180 - 144$

Step 1.2.2.2

Subtract

144

$144$ from

180

$180$ .

A = 36

$A = 36$

A = 36

$A = 36$

A = 36

$A = 36$

A = 36

$A = 36$

Step 2

Find

b

$b$ .

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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}

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

Step 2.2

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

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

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

Step 2.3

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

b

$b$ .

b = c \cdot cos (A)

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

Step 2.4

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

b = 27 \cdot cos (36)

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

Step 2.5

Multiply

27

$27$ by

0.80901699

$0.80901699$ .

b = 21.84345884

$b = 21.84345884$

b = 21.84345884

$b = 21.84345884$

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}

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

Step 3.2

Solve the equation for

a

$a$ .

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

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

Step 3.3

Substitute the actual values into the equation.

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

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

Step 3.4

Raise

27

$27$ to the power of

2

$2$ .

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

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

Step 3.5

Raise

21.84345884

$21.84345884$ to the power of

2

$2$ .

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

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

Step 3.6

Multiply

- 1

$- 1$ by

477.13669444

$477.13669444$ .

a = \sqrt{729 - 477.13669444}

$a = \sqrt{729 - 477.13669444}$

Step 3.7

Subtract

477.13669444

$477.13669444$ from

729

$729$ .

a = \sqrt{251.86330555}

$a = \sqrt{251.86330555}$

a = \sqrt{251.86330555}

$a = \sqrt{251.86330555}$

Step 4

Convert

\sqrt{251.86330555}

$\sqrt{251.86330555}$ to a decimal.

a = 15.87020181

$a = 15.87020181$

Step 5

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

A = 36

$A = 36$

B = 54

$B = 54$

C = 90

$C = 90$

a = 15.87020181

$a = 15.87020181$

b = 21.84345884

$b = 21.84345884$

c = 27

$c = 27$