Trigonometry Examples

B = 127

$B = 127$ ,

a = 32

$a = 32$ ,

C = 25

$C = 25$

Step 1

The sum of all the angles in a triangle is

180

$180$ degrees.

A + 25 + 127 = 180

$A + 25 + 127 = 180$

Step 2

Solve the equation for

A

$A$ .

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Step 2.1

Add

25

$25$ and

127

$127$ .

A + 152 = 180

$A + 152 = 180$

Step 2.2

Move all terms not containing

A

$A$ to the right side of the equation.

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Step 2.2.1

Subtract

152

$152$ from both sides of the equation.

A = 180 - 152

$A = 180 - 152$

Step 2.2.2

Subtract

152

$152$ from

180

$180$ .

A = 28

$A = 28$

A = 28

$A = 28$

A = 28

$A = 28$

Step 3

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 4

Substitute the known values into the law of sines to find

b

$b$ .

\frac{\sin (127)}{b} = \frac{\sin (28)}{32}

$\frac{\sin (127)}{b} = \frac{\sin (28)}{32}$

Step 5

Solve the equation for

b

$b$ .

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Step 5.1

Factor each term.

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Step 5.1.1

Evaluate

\sin (127)

$\sin (127)$ .

\frac{0.79863551}{b} = \frac{\sin (28)}{32}

$\frac{0.79863551}{b} = \frac{\sin (28)}{32}$

Step 5.1.2

Evaluate

\sin (28)

$\sin (28)$ .

\frac{0.79863551}{b} = \frac{0.46947156}{32}

$\frac{0.79863551}{b} = \frac{0.46947156}{32}$

Step 5.1.3

Divide

0.46947156

$0.46947156$ by

32

$32$ .

\frac{0.79863551}{b} = 0.01467098

$\frac{0.79863551}{b} = 0.01467098$

\frac{0.79863551}{b} = 0.01467098

$\frac{0.79863551}{b} = 0.01467098$

Step 5.2

Find the LCD of the terms in the equation.

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Step 5.2.1

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 5.2.2

The LCM of one and any expression is the expression.

b

$b$

b

$b$

Step 5.3

Multiply each term in

\frac{0.79863551}{b} = 0.01467098

$\frac{0.79863551}{b} = 0.01467098$ by

b

$b$ to eliminate the fractions.

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Step 5.3.1

Multiply each term in

\frac{0.79863551}{b} = 0.01467098

$\frac{0.79863551}{b} = 0.01467098$ by

b

$b$ .

\frac{0.79863551}{b} b = 0.01467098 b

$\frac{0.79863551}{b} b = 0.01467098 b$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of

b

$b$ .

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Step 5.3.2.1.1

Cancel the common factor.

\frac{0.79863551}{b} b = 0.01467098 b

$\frac{0.79863551}{b} b = 0.01467098 b$

Step 5.3.2.1.2

Rewrite the expression.

0.79863551 = 0.01467098 b

$0.79863551 = 0.01467098 b$

0.79863551 = 0.01467098 b

$0.79863551 = 0.01467098 b$

0.79863551 = 0.01467098 b

$0.79863551 = 0.01467098 b$

0.79863551 = 0.01467098 b

$0.79863551 = 0.01467098 b$

Step 5.4

Solve the equation.

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Step 5.4.1

Rewrite the equation as

0.01467098 b = 0.79863551

$0.01467098 b = 0.79863551$ .

0.01467098 b = 0.79863551

$0.01467098 b = 0.79863551$

Step 5.4.2

Divide each term in

0.01467098 b = 0.79863551

$0.01467098 b = 0.79863551$ by

0.01467098

$0.01467098$ and simplify.

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Step 5.4.2.1

Divide each term in

0.01467098 b = 0.79863551

$0.01467098 b = 0.79863551$ by

0.01467098

$0.01467098$ .

\frac{0.01467098 b}{0.01467098} = \frac{0.79863551}{0.01467098}

$\frac{0.01467098 b}{0.01467098} = \frac{0.79863551}{0.01467098}$

Step 5.4.2.2

Simplify the left side.

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Step 5.4.2.2.1

Cancel the common factor of

0.01467098

$0.01467098$ .

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Step 5.4.2.2.1.1

Cancel the common factor.

\frac{0.01467098 b}{0.01467098} = \frac{0.79863551}{0.01467098}

$\frac{0.01467098 b}{0.01467098} = \frac{0.79863551}{0.01467098}$

Step 5.4.2.2.1.2

Divide

b

$b$ by

1

$1$ .

b = \frac{0.79863551}{0.01467098}

$b = \frac{0.79863551}{0.01467098}$

b = \frac{0.79863551}{0.01467098}

$b = \frac{0.79863551}{0.01467098}$

b = \frac{0.79863551}{0.01467098}

$b = \frac{0.79863551}{0.01467098}$

Step 5.4.2.3

Simplify the right side.

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Step 5.4.2.3.1

Divide

0.79863551

$0.79863551$ by

0.01467098

$0.01467098$ .

b = 54.43638837

$b = 54.43638837$

b = 54.43638837

$b = 54.43638837$

b = 54.43638837

$b = 54.43638837$

b = 54.43638837

$b = 54.43638837$

b = 54.43638837

$b = 54.43638837$

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 (25)}{c} = \frac{\sin (28)}{32}

$\frac{\sin (25)}{c} = \frac{\sin (28)}{32}$

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 (25)

$\sin (25)$ .

\frac{0.42261826}{c} = \frac{\sin (28)}{32}

$\frac{0.42261826}{c} = \frac{\sin (28)}{32}$

Step 8.1.2

Evaluate

\sin (28)

$\sin (28)$ .

\frac{0.42261826}{c} = \frac{0.46947156}{32}

$\frac{0.42261826}{c} = \frac{0.46947156}{32}$

Step 8.1.3

Divide

0.46947156

$0.46947156$ by

32

$32$ .

\frac{0.42261826}{c} = 0.01467098

$\frac{0.42261826}{c} = 0.01467098$

\frac{0.42261826}{c} = 0.01467098

$\frac{0.42261826}{c} = 0.01467098$

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{0.42261826}{c} = 0.01467098

$\frac{0.42261826}{c} = 0.01467098$ by

c

$c$ to eliminate the fractions.

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Step 8.3.1

Multiply each term in

\frac{0.42261826}{c} = 0.01467098

$\frac{0.42261826}{c} = 0.01467098$ by

c

$c$ .

\frac{0.42261826}{c} c = 0.01467098 c

$\frac{0.42261826}{c} c = 0.01467098 c$

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{0.42261826}{c} c = 0.01467098 c

$\frac{0.42261826}{c} c = 0.01467098 c$

Step 8.3.2.1.2

Rewrite the expression.

0.42261826 = 0.01467098 c

$0.42261826 = 0.01467098 c$

0.42261826 = 0.01467098 c

$0.42261826 = 0.01467098 c$

0.42261826 = 0.01467098 c

$0.42261826 = 0.01467098 c$

0.42261826 = 0.01467098 c

$0.42261826 = 0.01467098 c$

Step 8.4

Solve the equation.

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Step 8.4.1

Rewrite the equation as

0.01467098 c = 0.42261826

$0.01467098 c = 0.42261826$ .

0.01467098 c = 0.42261826

$0.01467098 c = 0.42261826$

Step 8.4.2

Divide each term in

0.01467098 c = 0.42261826

$0.01467098 c = 0.42261826$ by

0.01467098

$0.01467098$ and simplify.

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Step 8.4.2.1

Divide each term in

0.01467098 c = 0.42261826

$0.01467098 c = 0.42261826$ by

0.01467098

$0.01467098$ .

\frac{0.01467098 c}{0.01467098} = \frac{0.42261826}{0.01467098}

$\frac{0.01467098 c}{0.01467098} = \frac{0.42261826}{0.01467098}$

Step 8.4.2.2

Simplify the left side.

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Step 8.4.2.2.1

Cancel the common factor of

0.01467098

$0.01467098$ .

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Step 8.4.2.2.1.1

Cancel the common factor.

\frac{0.01467098 c}{0.01467098} = \frac{0.42261826}{0.01467098}

$\frac{0.01467098 c}{0.01467098} = \frac{0.42261826}{0.01467098}$

Step 8.4.2.2.1.2

Divide

c

$c$ by

1

$1$ .

c = \frac{0.42261826}{0.01467098}

$c = \frac{0.42261826}{0.01467098}$

c = \frac{0.42261826}{0.01467098}

$c = \frac{0.42261826}{0.01467098}$

c = \frac{0.42261826}{0.01467098}

$c = \frac{0.42261826}{0.01467098}$

Step 8.4.2.3

Simplify the right side.

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Step 8.4.2.3.1

Divide

0.42261826

$0.42261826$ by

0.01467098

$0.01467098$ .

c = 28.80639733

$c = 28.80639733$

c = 28.80639733

$c = 28.80639733$

c = 28.80639733

$c = 28.80639733$

c = 28.80639733

$c = 28.80639733$

c = 28.80639733

$c = 28.80639733$

Step 9

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

A = 28

$A = 28$

B = 127

$B = 127$

C = 25

$C = 25$

a = 32

$a = 32$

b = 54.43638837

$b = 54.43638837$

c = 28.80639733

$c = 28.80639733$