Trigonometry Examples

a = 10

$a = 10$ ,

c = 8.9

$c = 8.9$ ,

A = 63

$A = 63$

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

C

$C$ .

\frac{\sin (C)}{8.9} = \frac{\sin (63)}{10}

$\frac{\sin (C)}{8.9} = \frac{\sin (63)}{10}$

Step 3

Solve the equation for

C

$C$ .

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Step 3.1

Multiply both sides of the equation by

8.9

$8.9$ .

8.9 \frac{\sin (C)}{8.9} = 8.9 \frac{\sin (63)}{10}

$8.9 \frac{\sin (C)}{8.9} = 8.9 \frac{\sin (63)}{10}$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Cancel the common factor of

8.9

$8.9$ .

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Step 3.2.1.1.1

Cancel the common factor.

8.9 \frac{\sin (C)}{8.9} = 8.9 \frac{\sin (63)}{10}

$8.9 \frac{\sin (C)}{8.9} = 8.9 \frac{\sin (63)}{10}$

Step 3.2.1.1.2

Rewrite the expression.

\sin (C) = 8.9 \frac{\sin (63)}{10}

$\sin (C) = 8.9 \frac{\sin (63)}{10}$

\sin (C) = 8.9 \frac{\sin (63)}{10}

$\sin (C) = 8.9 \frac{\sin (63)}{10}$

\sin (C) = 8.9 \frac{\sin (63)}{10}

$\sin (C) = 8.9 \frac{\sin (63)}{10}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify

8.9 \frac{\sin (63)}{10}

$8.9 \frac{\sin (63)}{10}$ .

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Step 3.2.2.1.1

Evaluate

\sin (63)

$\sin (63)$ .

\sin (C) = 8.9 (\frac{0.89100652}{10})

$\sin (C) = 8.9 (\frac{0.89100652}{10})$

Step 3.2.2.1.2

Divide

0.89100652

$0.89100652$ by

10

$10$ .

\sin (C) = 8.9 \cdot 0.08910065

$\sin (C) = 8.9 \cdot 0.08910065$

Step 3.2.2.1.3

Multiply

8.9

$8.9$ by

0.08910065

$0.08910065$ .

\sin (C) = 0.7929958

$\sin (C) = 0.7929958$

\sin (C) = 0.7929958

$\sin (C) = 0.7929958$

\sin (C) = 0.7929958

$\sin (C) = 0.7929958$

\sin (C) = 0.7929958

$\sin (C) = 0.7929958$

Step 3.3

Take the inverse sine of both sides of the equation to extract

C

$C$ from inside the sine.

C = \arcsin (0.7929958)

$C = \arcsin (0.7929958)$

Step 3.4

Simplify the right side.

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Step 3.4.1

Evaluate

\arcsin (0.7929958)

$\arcsin (0.7929958)$ .

C = 52.46636228

$C = 52.46636228$

C = 52.46636228

$C = 52.46636228$

Step 3.5

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from

180

$180$ to find the solution in the second quadrant.

C = 180 - 52.46636228

$C = 180 - 52.46636228$

Step 3.6

Subtract

52.46636228

$52.46636228$ from

180

$180$ .

C = 127.53363771

$C = 127.53363771$

Step 3.7

The solution to the equation

C = 52.46636228

$C = 52.46636228$ .

C = 52.46636228, 127.53363771

$C = 52.46636228, 127.53363771$

Step 3.8

Exclude the invalid angle.

C = 52.46636228

$C = 52.46636228$

C = 52.46636228

$C = 52.46636228$

Step 4

The sum of all the angles in a triangle is

180

$180$ degrees.

63 + 52.46636228 + B = 180

$63 + 52.46636228 + B = 180$

Step 5

Solve the equation for

B

$B$ .

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Step 5.1

Add

63

$63$ and

52.46636228

$52.46636228$ .

115.46636228 + B = 180

$115.46636228 + B = 180$

Step 5.2

Move all terms not containing

B

$B$ to the right side of the equation.

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Step 5.2.1

Subtract

115.46636228

$115.46636228$ from both sides of the equation.

B = 180 - 115.46636228

$B = 180 - 115.46636228$

Step 5.2.2

Subtract

115.46636228

$115.46636228$ from

180

$180$ .

B = 64.53363771

$B = 64.53363771$

B = 64.53363771

$B = 64.53363771$

B = 64.53363771

$B = 64.53363771$

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

b

$b$ .

\frac{\sin (64.53363771)}{b} = \frac{\sin (63)}{10}

$\frac{\sin (64.53363771)}{b} = \frac{\sin (63)}{10}$

Step 8

Solve the equation for

b

$b$ .

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Step 8.1

Factor each term.

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Step 8.1.1

Evaluate

\sin (64.53363771)

$\sin (64.53363771)$ .

\frac{0.90283787}{b} = \frac{\sin (63)}{10}

$\frac{0.90283787}{b} = \frac{\sin (63)}{10}$

Step 8.1.2

Evaluate

\sin (63)

$\sin (63)$ .

\frac{0.90283787}{b} = \frac{0.89100652}{10}

$\frac{0.90283787}{b} = \frac{0.89100652}{10}$

Step 8.1.3

Divide

0.89100652

$0.89100652$ by

10

$10$ .

\frac{0.90283787}{b} = 0.08910065

$\frac{0.90283787}{b} = 0.08910065$

\frac{0.90283787}{b} = 0.08910065

$\frac{0.90283787}{b} = 0.08910065$

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.

b, 1

$b, 1$

Step 8.2.2

The LCM of one and any expression is the expression.

b

$b$

b

$b$

Step 8.3

Multiply each term in

\frac{0.90283787}{b} = 0.08910065

$\frac{0.90283787}{b} = 0.08910065$ by

b

$b$ to eliminate the fractions.

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Step 8.3.1

Multiply each term in

\frac{0.90283787}{b} = 0.08910065

$\frac{0.90283787}{b} = 0.08910065$ by

b

$b$ .

\frac{0.90283787}{b} b = 0.08910065 b

$\frac{0.90283787}{b} b = 0.08910065 b$

Step 8.3.2

Simplify the left side.

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Step 8.3.2.1

Cancel the common factor of

b

$b$ .

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Step 8.3.2.1.1

Cancel the common factor.

\frac{0.90283787}{b} b = 0.08910065 b

$\frac{0.90283787}{b} b = 0.08910065 b$

Step 8.3.2.1.2

Rewrite the expression.

0.90283787 = 0.08910065 b

$0.90283787 = 0.08910065 b$

0.90283787 = 0.08910065 b

$0.90283787 = 0.08910065 b$

0.90283787 = 0.08910065 b

$0.90283787 = 0.08910065 b$

0.90283787 = 0.08910065 b

$0.90283787 = 0.08910065 b$

Step 8.4

Solve the equation.

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Step 8.4.1

Rewrite the equation as

0.08910065 b = 0.90283787

$0.08910065 b = 0.90283787$ .

0.08910065 b = 0.90283787

$0.08910065 b = 0.90283787$

Step 8.4.2

Divide each term in

0.08910065 b = 0.90283787

$0.08910065 b = 0.90283787$ by

0.08910065

$0.08910065$ and simplify.

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Step 8.4.2.1

Divide each term in

0.08910065 b = 0.90283787

$0.08910065 b = 0.90283787$ by

0.08910065

$0.08910065$ .

\frac{0.08910065 b}{0.08910065} = \frac{0.90283787}{0.08910065}

$\frac{0.08910065 b}{0.08910065} = \frac{0.90283787}{0.08910065}$

Step 8.4.2.2

Simplify the left side.

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Step 8.4.2.2.1

Cancel the common factor of

0.08910065

$0.08910065$ .

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Step 8.4.2.2.1.1

Cancel the common factor.

\frac{0.08910065 b}{0.08910065} = \frac{0.90283787}{0.08910065}

$\frac{0.08910065 b}{0.08910065} = \frac{0.90283787}{0.08910065}$

Step 8.4.2.2.1.2

Divide

b

$b$ by

1

$1$ .

b = \frac{0.90283787}{0.08910065}

$b = \frac{0.90283787}{0.08910065}$

b = \frac{0.90283787}{0.08910065}

$b = \frac{0.90283787}{0.08910065}$

b = \frac{0.90283787}{0.08910065}

$b = \frac{0.90283787}{0.08910065}$

Step 8.4.2.3

Simplify the right side.

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Step 8.4.2.3.1

Divide

0.90283787

$0.90283787$ by

0.08910065

$0.08910065$ .

b = 10.13278637

$b = 10.13278637$

b = 10.13278637

$b = 10.13278637$

b = 10.13278637

$b = 10.13278637$

b = 10.13278637

$b = 10.13278637$

b = 10.13278637

$b = 10.13278637$

Step 9

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

A = 63

$A = 63$

B = 64.53363771

$B = 64.53363771$

C = 52.46636228

$C = 52.46636228$

a = 10

$a = 10$

b = 10.13278637

$b = 10.13278637$

c = 8.9

$c = 8.9$