Trigonometry Examples



\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = \\ a = & 105 \end{matrix} & \begin{matrix} A = & 65 \\ B = & 37 \\ C = \end{matrix} \end{matrix}

$\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = \\ a = & 105 \end{matrix} & \begin{matrix} A = & 65 \\ B = & 37 \\ C = \end{matrix} \end{matrix}$

Step 1

The law of sines is based on the proportionality of sides and angles in triangles. The law states that for the angles of a non-right triangle, each angle of the triangle has the same ratio of angle measure to sine value.

\frac{\sin (A)}{a} = \frac{\sin (B)}{b} = \frac{\sin (C)}{c}

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

Step 2

Substitute the known values into the law of sines to find

b

$b$ .

\frac{\sin (37)}{b} = \frac{\sin (65)}{105}

$\frac{\sin (37)}{b} = \frac{\sin (65)}{105}$

Step 3

Solve the equation for

b

$b$ .

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Step 3.1

Factor each term.

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Step 3.1.1

Evaluate

\sin (37)

$\sin (37)$ .

\frac{0.60181502}{b} = \frac{\sin (65)}{105}

$\frac{0.60181502}{b} = \frac{\sin (65)}{105}$

Step 3.1.2

Evaluate

\sin (65)

$\sin (65)$ .

\frac{0.60181502}{b} = \frac{0.90630778}{105}

$\frac{0.60181502}{b} = \frac{0.90630778}{105}$

Step 3.1.3

Divide

0.90630778

$0.90630778$ by

105

$105$ .

\frac{0.60181502}{b} = 0.0086315

$\frac{0.60181502}{b} = 0.0086315$

\frac{0.60181502}{b} = 0.0086315

$\frac{0.60181502}{b} = 0.0086315$

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

b, 1

$b, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

b

$b$

b

$b$

Step 3.3

Multiply each term in

\frac{0.60181502}{b} = 0.0086315

$\frac{0.60181502}{b} = 0.0086315$ by

b

$b$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in

\frac{0.60181502}{b} = 0.0086315

$\frac{0.60181502}{b} = 0.0086315$ by

b

$b$ .

\frac{0.60181502}{b} b = 0.0086315 b

$\frac{0.60181502}{b} b = 0.0086315 b$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of

b

$b$ .

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Step 3.3.2.1.1

Cancel the common factor.

\frac{0.60181502}{b} b = 0.0086315 b

$\frac{0.60181502}{b} b = 0.0086315 b$

Step 3.3.2.1.2

Rewrite the expression.

0.60181502 = 0.0086315 b

$0.60181502 = 0.0086315 b$

0.60181502 = 0.0086315 b

$0.60181502 = 0.0086315 b$

0.60181502 = 0.0086315 b

$0.60181502 = 0.0086315 b$

0.60181502 = 0.0086315 b

$0.60181502 = 0.0086315 b$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as

0.0086315 b = 0.60181502

$0.0086315 b = 0.60181502$ .

0.0086315 b = 0.60181502

$0.0086315 b = 0.60181502$

Step 3.4.2

Divide each term in

0.0086315 b = 0.60181502

$0.0086315 b = 0.60181502$ by

0.0086315

$0.0086315$ and simplify.

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Step 3.4.2.1

Divide each term in

0.0086315 b = 0.60181502

$0.0086315 b = 0.60181502$ by

0.0086315

$0.0086315$ .

\frac{0.0086315 b}{0.0086315} = \frac{0.60181502}{0.0086315}

$\frac{0.0086315 b}{0.0086315} = \frac{0.60181502}{0.0086315}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of

0.0086315

$0.0086315$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

\frac{0.0086315 b}{0.0086315} = \frac{0.60181502}{0.0086315}

$\frac{0.0086315 b}{0.0086315} = \frac{0.60181502}{0.0086315}$

Step 3.4.2.2.1.2

Divide

b

$b$ by

1

$1$ .

b = \frac{0.60181502}{0.0086315}

$b = \frac{0.60181502}{0.0086315}$

b = \frac{0.60181502}{0.0086315}

$b = \frac{0.60181502}{0.0086315}$

b = \frac{0.60181502}{0.0086315}

$b = \frac{0.60181502}{0.0086315}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Divide

0.60181502

$0.60181502$ by

0.0086315

$0.0086315$ .

b = 69.72308782

$b = 69.72308782$

b = 69.72308782

$b = 69.72308782$

b = 69.72308782

$b = 69.72308782$

b = 69.72308782

$b = 69.72308782$

b = 69.72308782

$b = 69.72308782$

Step 4

The sum of all the angles in a triangle is

180

$180$ degrees.

65 + C + 37 = 180

$65 + C + 37 = 180$

Step 5

Solve the equation for

C

$C$ .

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Step 5.1

Add

65

$65$ and

37

$37$ .

C + 102 = 180

$C + 102 = 180$

Step 5.2

Move all terms not containing

C

$C$ to the right side of the equation.

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Step 5.2.1

Subtract

102

$102$ from both sides of the equation.

C = 180 - 102

$C = 180 - 102$

Step 5.2.2

Subtract

102

$102$ from

180

$180$ .

C = 78

$C = 78$

C = 78

$C = 78$

C = 78

$C = 78$

Step 6

\frac{\sin (A)}{a} = \frac{\sin (B)}{b} = \frac{\sin (C)}{c}

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

Step 7

Substitute the known values into the law of sines to find

c

$c$ .

\frac{\sin (78)}{c} = \frac{\sin (65)}{105}

$\frac{\sin (78)}{c} = \frac{\sin (65)}{105}$

Step 8

Solve the equation for

c

$c$ .

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Step 8.1

Factor each term.

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Step 8.1.1

Evaluate

\sin (78)

$\sin (78)$ .

\frac{0.9781476}{c} = \frac{\sin (65)}{105}

$\frac{0.9781476}{c} = \frac{\sin (65)}{105}$

Step 8.1.2

Evaluate

\sin (65)

$\sin (65)$ .

\frac{0.9781476}{c} = \frac{0.90630778}{105}

$\frac{0.9781476}{c} = \frac{0.90630778}{105}$

Step 8.1.3

Divide

0.90630778

$0.90630778$ by

105

$105$ .

\frac{0.9781476}{c} = 0.0086315

$\frac{0.9781476}{c} = 0.0086315$

\frac{0.9781476}{c} = 0.0086315

$\frac{0.9781476}{c} = 0.0086315$

Step 8.2

Find the LCD of the terms in the equation.

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Step 8.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

c, 1

$c, 1$

Step 8.2.2

The LCM of one and any expression is the expression.

c

$c$

c

$c$

Step 8.3

Multiply each term in

\frac{0.9781476}{c} = 0.0086315

$\frac{0.9781476}{c} = 0.0086315$ by

c

$c$ to eliminate the fractions.

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Step 8.3.1

Multiply each term in

\frac{0.9781476}{c} = 0.0086315

$\frac{0.9781476}{c} = 0.0086315$ by

c

$c$ .

\frac{0.9781476}{c} c = 0.0086315 c

$\frac{0.9781476}{c} c = 0.0086315 c$

Step 8.3.2

Simplify the left side.

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Step 8.3.2.1

Cancel the common factor of

c

$c$ .

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Step 8.3.2.1.1

Cancel the common factor.

\frac{0.9781476}{c} c = 0.0086315 c

$\frac{0.9781476}{c} c = 0.0086315 c$

Step 8.3.2.1.2

Rewrite the expression.

0.9781476 = 0.0086315 c

$0.9781476 = 0.0086315 c$

0.9781476 = 0.0086315 c

$0.9781476 = 0.0086315 c$

0.9781476 = 0.0086315 c

$0.9781476 = 0.0086315 c$

0.9781476 = 0.0086315 c

$0.9781476 = 0.0086315 c$

Step 8.4

Solve the equation.

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Step 8.4.1

Rewrite the equation as

0.0086315 c = 0.9781476

$0.0086315 c = 0.9781476$ .

0.0086315 c = 0.9781476

$0.0086315 c = 0.9781476$

Step 8.4.2

Divide each term in

0.0086315 c = 0.9781476

$0.0086315 c = 0.9781476$ by

0.0086315

$0.0086315$ and simplify.

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Step 8.4.2.1

Divide each term in

0.0086315 c = 0.9781476

$0.0086315 c = 0.9781476$ by

0.0086315

$0.0086315$ .

\frac{0.0086315 c}{0.0086315} = \frac{0.9781476}{0.0086315}

$\frac{0.0086315 c}{0.0086315} = \frac{0.9781476}{0.0086315}$

Step 8.4.2.2

Simplify the left side.

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Step 8.4.2.2.1

Cancel the common factor of

0.0086315

$0.0086315$ .

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Step 8.4.2.2.1.1

Cancel the common factor.

\frac{0.0086315 c}{0.0086315} = \frac{0.9781476}{0.0086315}

$\frac{0.0086315 c}{0.0086315} = \frac{0.9781476}{0.0086315}$

Step 8.4.2.2.1.2

Divide

c

$c$ by

1

$1$ .

c = \frac{0.9781476}{0.0086315}

$c = \frac{0.9781476}{0.0086315}$

c = \frac{0.9781476}{0.0086315}

$c = \frac{0.9781476}{0.0086315}$

c = \frac{0.9781476}{0.0086315}

$c = \frac{0.9781476}{0.0086315}$

Step 8.4.2.3

Simplify the right side.

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Step 8.4.2.3.1

Divide

0.9781476

$0.9781476$ by

0.0086315

$0.0086315$ .

c = 113.32297873

$c = 113.32297873$

c = 113.32297873

$c = 113.32297873$

c = 113.32297873

$c = 113.32297873$

c = 113.32297873

$c = 113.32297873$

c = 113.32297873

$c = 113.32297873$

Step 9

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

A = 65

$A = 65$

B = 37

$B = 37$

C = 78

$C = 78$

a = 105

$a = 105$

b = 69.72308782

$b = 69.72308782$

c = 113.32297873

$c = 113.32297873$