Trigonometry Examples

A = 76

$A = 76$ ,

B = 34

$B = 34$ ,

c = 9

$c = 9$

Step 1

The sum of all the angles in a triangle is

180

$180$ degrees.

76 + C + 34 = 180

$76 + C + 34 = 180$

Step 2

Solve the equation for

C

$C$ .

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Step 2.1

Add

76

$76$ and

34

$34$ .

C + 110 = 180

$C + 110 = 180$

Step 2.2

Move all terms not containing

C

$C$ to the right side of the equation.

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Step 2.2.1

Subtract

110

$110$ from both sides of the equation.

C = 180 - 110

$C = 180 - 110$

Step 2.2.2

Subtract

110

$110$ from

180

$180$ .

C = 70

$C = 70$

C = 70

$C = 70$

C = 70

$C = 70$

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

a

$a$ .

\frac{\sin (76)}{a} = \frac{\sin (70)}{9}

$\frac{\sin (76)}{a} = \frac{\sin (70)}{9}$

Step 5

Solve the equation for

a

$a$ .

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Step 5.1

Factor each term.

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Step 5.1.1

Evaluate

\sin (76)

$\sin (76)$ .

\frac{0.97029572}{a} = \frac{\sin (70)}{9}

$\frac{0.97029572}{a} = \frac{\sin (70)}{9}$

Step 5.1.2

Evaluate

\sin (70)

$\sin (70)$ .

\frac{0.97029572}{a} = \frac{0.93969262}{9}

$\frac{0.97029572}{a} = \frac{0.93969262}{9}$

Step 5.1.3

Divide

0.93969262

$0.93969262$ by

9

$9$ .

\frac{0.97029572}{a} = 0.10441029

$\frac{0.97029572}{a} = 0.10441029$

\frac{0.97029572}{a} = 0.10441029

$\frac{0.97029572}{a} = 0.10441029$

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.

a, 1

$a, 1$

Step 5.2.2

The LCM of one and any expression is the expression.

a

$a$

a

$a$

Step 5.3

Multiply each term in

\frac{0.97029572}{a} = 0.10441029

$\frac{0.97029572}{a} = 0.10441029$ by

a

$a$ to eliminate the fractions.

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Step 5.3.1

Multiply each term in

\frac{0.97029572}{a} = 0.10441029

$\frac{0.97029572}{a} = 0.10441029$ by

a

$a$ .

\frac{0.97029572}{a} a = 0.10441029 a

$\frac{0.97029572}{a} a = 0.10441029 a$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of

a

$a$ .

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Step 5.3.2.1.1

Cancel the common factor.

\frac{0.97029572}{a} a = 0.10441029 a

$\frac{0.97029572}{a} a = 0.10441029 a$

Step 5.3.2.1.2

Rewrite the expression.

0.97029572 = 0.10441029 a

$0.97029572 = 0.10441029 a$

0.97029572 = 0.10441029 a

$0.97029572 = 0.10441029 a$

0.97029572 = 0.10441029 a

$0.97029572 = 0.10441029 a$

0.97029572 = 0.10441029 a

$0.97029572 = 0.10441029 a$

Step 5.4

Solve the equation.

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Step 5.4.1

Rewrite the equation as

0.10441029 a = 0.97029572

$0.10441029 a = 0.97029572$ .

0.10441029 a = 0.97029572

$0.10441029 a = 0.97029572$

Step 5.4.2

Divide each term in

0.10441029 a = 0.97029572

$0.10441029 a = 0.97029572$ by

0.10441029

$0.10441029$ and simplify.

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Step 5.4.2.1

Divide each term in

0.10441029 a = 0.97029572

$0.10441029 a = 0.97029572$ by

0.10441029

$0.10441029$ .

\frac{0.10441029 a}{0.10441029} = \frac{0.97029572}{0.10441029}

$\frac{0.10441029 a}{0.10441029} = \frac{0.97029572}{0.10441029}$

Step 5.4.2.2

Simplify the left side.

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Step 5.4.2.2.1

Cancel the common factor of

0.10441029

$0.10441029$ .

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Step 5.4.2.2.1.1

Cancel the common factor.

\frac{0.10441029 a}{0.10441029} = \frac{0.97029572}{0.10441029}

$\frac{0.10441029 a}{0.10441029} = \frac{0.97029572}{0.10441029}$

Step 5.4.2.2.1.2

Divide

a

$a$ by

1

$1$ .

a = \frac{0.97029572}{0.10441029}

$a = \frac{0.97029572}{0.10441029}$

a = \frac{0.97029572}{0.10441029}

$a = \frac{0.97029572}{0.10441029}$

a = \frac{0.97029572}{0.10441029}

$a = \frac{0.97029572}{0.10441029}$

Step 5.4.2.3

Simplify the right side.

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Step 5.4.2.3.1

Divide

0.97029572

$0.97029572$ by

0.10441029

$0.10441029$ .

a = 9.2931043

$a = 9.2931043$

a = 9.2931043

$a = 9.2931043$

a = 9.2931043

$a = 9.2931043$

a = 9.2931043

$a = 9.2931043$

a = 9.2931043

$a = 9.2931043$

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 (34)}{b} = \frac{\sin (76)}{9.2931043}

$\frac{\sin (34)}{b} = \frac{\sin (76)}{9.2931043}$

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 (34)

$\sin (34)$ .

\frac{0.5591929}{b} = \frac{\sin (76)}{9.2931043}

$\frac{0.5591929}{b} = \frac{\sin (76)}{9.2931043}$

Step 8.1.2

Evaluate

\sin (76)

$\sin (76)$ .

\frac{0.5591929}{b} = \frac{0.97029572}{9.2931043}

$\frac{0.5591929}{b} = \frac{0.97029572}{9.2931043}$

Step 8.1.3

Divide

0.97029572

$0.97029572$ by

9.2931043

$9.2931043$ .

\frac{0.5591929}{b} = 0.10441029

$\frac{0.5591929}{b} = 0.10441029$

\frac{0.5591929}{b} = 0.10441029

$\frac{0.5591929}{b} = 0.10441029$

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{0.5591929}{b} = 0.10441029

$\frac{0.5591929}{b} = 0.10441029$ by

b

$b$ to eliminate the fractions.

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Step 8.3.1

Multiply each term in

\frac{0.5591929}{b} = 0.10441029

$\frac{0.5591929}{b} = 0.10441029$ by

b

$b$ .

\frac{0.5591929}{b} b = 0.10441029 b

$\frac{0.5591929}{b} b = 0.10441029 b$

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{0.5591929}{b} b = 0.10441029 b

$\frac{0.5591929}{b} b = 0.10441029 b$

Step 8.3.2.1.2

Rewrite the expression.

0.5591929 = 0.10441029 b

$0.5591929 = 0.10441029 b$

0.5591929 = 0.10441029 b

$0.5591929 = 0.10441029 b$

0.5591929 = 0.10441029 b

$0.5591929 = 0.10441029 b$

0.5591929 = 0.10441029 b

$0.5591929 = 0.10441029 b$

Step 8.4

Solve the equation.

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Step 8.4.1

Rewrite the equation as

0.10441029 b = 0.5591929

$0.10441029 b = 0.5591929$ .

0.10441029 b = 0.5591929

$0.10441029 b = 0.5591929$

Step 8.4.2

Divide each term in

0.10441029 b = 0.5591929

$0.10441029 b = 0.5591929$ by

0.10441029

$0.10441029$ and simplify.

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Step 8.4.2.1

Divide each term in

0.10441029 b = 0.5591929

$0.10441029 b = 0.5591929$ by

0.10441029

$0.10441029$ .

\frac{0.10441029 b}{0.10441029} = \frac{0.5591929}{0.10441029}

$\frac{0.10441029 b}{0.10441029} = \frac{0.5591929}{0.10441029}$

Step 8.4.2.2

Simplify the left side.

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Step 8.4.2.2.1

Cancel the common factor of

0.10441029

$0.10441029$ .

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Step 8.4.2.2.1.1

Cancel the common factor.

\frac{0.10441029 b}{0.10441029} = \frac{0.5591929}{0.10441029}

$\frac{0.10441029 b}{0.10441029} = \frac{0.5591929}{0.10441029}$

Step 8.4.2.2.1.2

Divide

b

$b$ by

1

$1$ .

b = \frac{0.5591929}{0.10441029}

$b = \frac{0.5591929}{0.10441029}$

b = \frac{0.5591929}{0.10441029}

$b = \frac{0.5591929}{0.10441029}$

b = \frac{0.5591929}{0.10441029}

$b = \frac{0.5591929}{0.10441029}$

Step 8.4.2.3

Simplify the right side.

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Step 8.4.2.3.1

Divide

0.5591929

$0.5591929$ by

0.10441029

$0.10441029$ .

b = 5.35572592

$b = 5.35572592$

b = 5.35572592

$b = 5.35572592$

b = 5.35572592

$b = 5.35572592$

b = 5.35572592

$b = 5.35572592$

b = 5.35572592

$b = 5.35572592$

Step 9

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

A = 76

$A = 76$

B = 34

$B = 34$

C = 70

$C = 70$

a = 9.2931043

$a = 9.2931043$

b = 5.35572592

$b = 5.35572592$

c = 9

$c = 9$

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