Trigonometry Examples

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\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = \\ a = & 8 \end{matrix} & \begin{matrix} A = \\ B = & 40 \\ C = & 90 \end{matrix} \end{matrix}

$\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = \\ a = & 8 \end{matrix} & \begin{matrix} A = \\ B = & 40 \\ C = & 90 \end{matrix} \end{matrix}$

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 + 40 = 180

$A + 90 + 40 = 180$

Step 1.2

Solve the equation for

A

$A$ .

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Step 1.2.1

Add

90

$90$ and

40

$40$ .

A + 130 = 180

$A + 130 = 180$

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

130

$130$ from both sides of the equation.

A = 180 - 130

$A = 180 - 130$

Step 1.2.2.2

Subtract

130

$130$ from

180

$180$ .

A = 50

$A = 50$

A = 50

$A = 50$

A = 50

$A = 50$

A = 50

$A = 50$

Step 2

Find

c

$c$ .

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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 (B) = \frac{adj}{hyp}

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

Step 2.2

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

\cos (B) = \frac{a}{c}

$\cos (B) = \frac{a}{c}$

Step 2.3

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

c

$c$ .

c = \frac{a}{\cos (B)}

$c = \frac{a}{\cos (B)}$

Step 2.4

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

c = \frac{8}{\cos (40)}

$c = \frac{8}{\cos (40)}$

Step 2.5

The value of

\cos (40)

$\cos (40)$ is

0.76604444

$0.76604444$ .

c = \frac{8}{0.76604444}

$c = \frac{8}{0.76604444}$

Step 2.6

Divide

8

$8$ by

0.76604444

$0.76604444$ .

c = 10.44325831

$c = 10.44325831$

c = 10.44325831

$c = 10.44325831$

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

b

$b$ .

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

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

Step 3.3

Substitute the actual values into the equation.

b = \sqrt{{(10.44325831)}^{2} - {(8)}^{2}}

$b = \sqrt{{(10.44325831)}^{2} - {(8)}^{2}}$

Step 3.4

Raise

10.44325831

$10.44325831$ to the power of

2

$2$ .

b = \sqrt{109.06164422 - {(8)}^{2}}

$b = \sqrt{109.06164422 - {(8)}^{2}}$

Step 3.5

Raise

8

$8$ to the power of

2

$2$ .

b = \sqrt{109.06164422 - 1 \cdot 64}

$b = \sqrt{109.06164422 - 1 \cdot 64}$

Step 3.6

Multiply

- 1

$- 1$ by

64

$64$ .

b = \sqrt{109.06164422 - 64}

$b = \sqrt{109.06164422 - 64}$

Step 3.7

Subtract

64

$64$ from

109.06164422

$109.06164422$ .

b = \sqrt{45.06164422}

$b = \sqrt{45.06164422}$

b = \sqrt{45.06164422}

$b = \sqrt{45.06164422}$

Step 4

Convert

\sqrt{45.06164422}

$\sqrt{45.06164422}$ to a decimal.

b = 6.71279704

$b = 6.71279704$

Step 5

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

A = 50

$A = 50$

B = 40

$B = 40$

C = 90

$C = 90$

a = 8

$a = 8$

b = 6.71279704

$b = 6.71279704$

c = 10.44325831

$c = 10.44325831$