Trigonometry Examples

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\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = & 4 \\ a = \end{matrix} & \begin{matrix} A = \\ B = & 70 \\ C = & 90 \end{matrix} \end{matrix}

$\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = & 4 \\ a = \end{matrix} & \begin{matrix} A = \\ B = & 70 \\ C = & 90 \end{matrix} \end{matrix}$

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 + 70 = 180

$A + 90 + 70 = 180$

Step 1.2

Solve the equation for

A

$A$ .

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Step 1.2.1

Add

90

$90$ and

70

$70$ .

A + 160 = 180

$A + 160 = 180$

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

160

$160$ from both sides of the equation.

A = 180 - 160

$A = 180 - 160$

Step 1.2.2.2

Subtract

160

$160$ from

180

$180$ .

A = 20

$A = 20$

A = 20

$A = 20$

A = 20

$A = 20$

A = 20

$A = 20$

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 = 4 \cdot \cos (20)

$b = 4 \cdot \cos (20)$

Step 2.5

Multiply

4

$4$ by

0.93969262

$0.93969262$ .

b = 3.75877048

$b = 3.75877048$

b = 3.75877048

$b = 3.75877048$

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{{(4)}^{2} - {(3.75877048)}^{2}}

$a = \sqrt{{(4)}^{2} - {(3.75877048)}^{2}}$

Step 3.4

Raise

4

$4$ to the power of

2

$2$ .

a = \sqrt{16 - {(3.75877048)}^{2}}

$a = \sqrt{16 - {(3.75877048)}^{2}}$

Step 3.5

Raise

3.75877048

$3.75877048$ to the power of

2

$2$ .

a = \sqrt{16 - 1 \cdot 14.12835554}

$a = \sqrt{16 - 1 \cdot 14.12835554}$

Step 3.6

Multiply

- 1

$- 1$ by

14.12835554

$14.12835554$ .

a = \sqrt{16 - 14.12835554}

$a = \sqrt{16 - 14.12835554}$

Step 3.7

Subtract

14.12835554

$14.12835554$ from

16

$16$ .

a = \sqrt{1.87164445}

$a = \sqrt{1.87164445}$

a = \sqrt{1.87164445}

$a = \sqrt{1.87164445}$

Step 4

Convert

\sqrt{1.87164445}

$\sqrt{1.87164445}$ to a decimal.

a = 1.36808057

$a = 1.36808057$

Step 5

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

A = 20

$A = 20$

B = 70

$B = 70$

C = 90

$C = 90$

a = 1.36808057

$a = 1.36808057$

b = 3.75877048

$b = 3.75877048$

c = 4

$c = 4$