Trigonometry Examples

$B = 32$ , $b = 100$ , $c = 150$

Step 1

The Law of Sines produces an ambiguous angle result. This means that there are $2$ angles that will correctly solve the equation. For the first triangle, use the first possible angle value.

Solve for the first triangle.

Step 2

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.

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

Step 3

Substitute the known values into the law of sines to find $C$ .

$\frac{\sin (C)}{150} = \frac{\sin (32)}{100}$

Step 4

Solve the equation for $C$ .

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Step 4.1

Multiply both sides of the equation by $150$ .

$150 \frac{\sin (C)}{150} = 150 \frac{\sin (32)}{100}$

Step 4.2

Simplify both sides of the equation.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Cancel the common factor of $150$ .

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Step 4.2.1.1.1

Cancel the common factor.

$150 \frac{\sin (C)}{150} = 150 \frac{\sin (32)}{100}$

Step 4.2.1.1.2

Rewrite the expression.

$\sin (C) = 150 \frac{\sin (32)}{100}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $150 \frac{\sin (32)}{100}$ .

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Step 4.2.2.1.1

Cancel the common factor of $50$ .

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Step 4.2.2.1.1.1

Factor $50$ out of $150$ .

$\sin (C) = 50 (3) \frac{\sin (32)}{100}$

Step 4.2.2.1.1.2

Factor $50$ out of $100$ .

$\sin (C) = 50 \cdot 3 \frac{\sin (32)}{50 \cdot 2}$

Step 4.2.2.1.1.3

Cancel the common factor.

$\sin (C) = 50 \cdot 3 \frac{\sin (32)}{50 \cdot 2}$

Step 4.2.2.1.1.4

Rewrite the expression.

$\sin (C) = 3 \frac{\sin (32)}{2}$

Step 4.2.2.1.2

Combine $3$ and $\frac{\sin (32)}{2}$ .

$\sin (C) = \frac{3 \sin (32)}{2}$

Step 4.2.2.1.3

Evaluate $\sin (32)$ .

$\sin (C) = \frac{3 \cdot 0.52991926}{2}$

Step 4.2.2.1.4

Multiply $3$ by $0.52991926$ .

$\sin (C) = \frac{1.58975779}{2}$

Step 4.2.2.1.5

Divide $1.58975779$ by $2$ .

$\sin (C) = 0.79487889$

Step 4.3

Take the inverse sine of both sides of the equation to extract $C$ from inside the sine.

$C = \arcsin (0.79487889)$

Step 4.4

Simplify the right side.

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Step 4.4.1

Evaluate $\arcsin (0.79487889)$ .

$C = 52.64381859$

Step 4.5

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the second quadrant.

$C = 180 - 52.64381859$

Step 4.6

Subtract $52.64381859$ from $180$ .

$C = 127.3561814$

Step 4.7

The solution to the equation $C = 52.64381859$ .

$C = 52.64381859, 127.3561814$

Step 5

The sum of all the angles in a triangle is $180$ degrees.

$A + 52.64381859 + 32 = 180$

Step 6

Solve the equation for $A$ .

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Step 6.1

Add $52.64381859$ and $32$ .

$A + 84.64381859 = 180$

Step 6.2

Move all terms not containing $A$ to the right side of the equation.

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Step 6.2.1

Subtract $84.64381859$ from both sides of the equation.

$A = 180 - 84.64381859$

Step 6.2.2

Subtract $84.64381859$ from $180$ .

$A = 95.3561814$

Step 7

Use the law of cosines to find the unknown side of the triangle, given the other two sides and the included angle.

$a^{2} = b^{2} + c^{2} - 2 b c \cos (A)$

Step 8

Solve the equation.

$a = \sqrt{b^{2} + c^{2} - 2 b c \cos (A)}$

Step 9

Substitute the known values into the equation.

$a = \sqrt{{(100)}^{2} + {(150)}^{2} - 2 \cdot 100 \cdot 150 \cos (95.3561814)}$

Step 10

Simplify the results.

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Step 10.1

Raise $100$ to the power of $2$ .

$a = \sqrt{10000 + {(150)}^{2} - 2 \cdot 100 \cdot (150 \cos (95.3561814))}$

Step 10.2

Raise $150$ to the power of $2$ .

$a = \sqrt{10000 + 22500 - 2 \cdot 100 \cdot (150 \cos (95.3561814))}$

Step 10.3

Multiply $- 2 \cdot 100 \cdot 150$ .

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Step 10.3.1

Multiply $- 2$ by $100$ .

$a = \sqrt{10000 + 22500 - 200 \cdot (150 \cos (95.3561814))}$

Step 10.3.2

Multiply $- 200$ by $150$ .

$a = \sqrt{10000 + 22500 - 30000 \cos (95.3561814)}$

Step 10.4

Add $10000$ and $22500$ .

$a = \sqrt{32500 - 30000 \cos (95.3561814)}$

Step 10.5

Rewrite $32500 - 30000 \cos (95.3561814)$ as $50^{2} (13 - 12 \cos (95.3561814))$ .

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Step 10.5.1

Factor $2500$ out of $32500$ .

$a = \sqrt{2500 (13) - 30000 \cos (95.3561814)}$

Step 10.5.2

Factor $2500$ out of $- 30000 \cos (95.3561814)$ .

$a = \sqrt{2500 (13) + 2500 (- 12 \cos (95.3561814))}$

Step 10.5.3

Factor $2500$ out of $2500 (13) + 2500 (- 12 \cos (95.3561814))$ .

$a = \sqrt{2500 (13 - 12 \cos (95.3561814))}$

Step 10.5.4

Rewrite $2500$ as $50^{2}$ .

$a = \sqrt{50^{2} (13 - 12 \cos (95.3561814))}$

Step 10.6

Pull terms out from under the radical.

$a = 50 \sqrt{13 - 12 \cos (95.3561814)}$

Step 11

For the second triangle, use the second possible angle value.

Solve for the second triangle.

Step 12

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

Step 13

Substitute the known values into the law of sines to find $C$ .

$\frac{\sin (C)}{150} = \frac{\sin (32)}{100}$

Step 14

Solve the equation for $C$ .

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Step 14.1

Multiply both sides of the equation by $150$ .

$150 \frac{\sin (C)}{150} = 150 \frac{\sin (32)}{100}$

Step 14.2

Simplify both sides of the equation.

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Step 14.2.1

Simplify the left side.

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Step 14.2.1.1

Cancel the common factor of $150$ .

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Step 14.2.1.1.1

Cancel the common factor.

$150 \frac{\sin (C)}{150} = 150 \frac{\sin (32)}{100}$

Step 14.2.1.1.2

Rewrite the expression.

$\sin (C) = 150 \frac{\sin (32)}{100}$

Step 14.2.2

Simplify the right side.

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Step 14.2.2.1

Simplify $150 \frac{\sin (32)}{100}$ .

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Step 14.2.2.1.1

Cancel the common factor of $50$ .

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Step 14.2.2.1.1.1

Factor $50$ out of $150$ .

$\sin (C) = 50 (3) \frac{\sin (32)}{100}$

Step 14.2.2.1.1.2

Factor $50$ out of $100$ .

$\sin (C) = 50 \cdot 3 \frac{\sin (32)}{50 \cdot 2}$

Step 14.2.2.1.1.3

Cancel the common factor.

$\sin (C) = 50 \cdot 3 \frac{\sin (32)}{50 \cdot 2}$

Step 14.2.2.1.1.4

Rewrite the expression.

$\sin (C) = 3 \frac{\sin (32)}{2}$

Step 14.2.2.1.2

Combine $3$ and $\frac{\sin (32)}{2}$ .

$\sin (C) = \frac{3 \sin (32)}{2}$

Step 14.2.2.1.3

Evaluate $\sin (32)$ .

$\sin (C) = \frac{3 \cdot 0.52991926}{2}$

Step 14.2.2.1.4

Multiply $3$ by $0.52991926$ .

$\sin (C) = \frac{1.58975779}{2}$

Step 14.2.2.1.5

Divide $1.58975779$ by $2$ .

$\sin (C) = 0.79487889$

Step 14.3

Take the inverse sine of both sides of the equation to extract $C$ from inside the sine.

$C = \arcsin (0.79487889)$

Step 14.4

Simplify the right side.

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Step 14.4.1

Evaluate $\arcsin (0.79487889)$ .

$C = 52.64381859$

Step 14.5

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the second quadrant.

$C = 180 - 52.64381859$

Step 14.6

Subtract $52.64381859$ from $180$ .

$C = 127.3561814$

Step 14.7

The solution to the equation $C = 52.64381859$ .

$C = 52.64381859, 127.3561814$

Step 15

The sum of all the angles in a triangle is $180$ degrees.

$A + 127.3561814 + 32 = 180$

Step 16

Solve the equation for $A$ .

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Step 16.1

Add $127.3561814$ and $32$ .

$A + 159.3561814 = 180$

Step 16.2

Move all terms not containing $A$ to the right side of the equation.

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Step 16.2.1

Subtract $159.3561814$ from both sides of the equation.

$A = 180 - 159.3561814$

Step 16.2.2

Subtract $159.3561814$ from $180$ .

$A = 20.64381859$

Step 17

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

Step 18

Substitute the known values into the law of sines to find $a$ .

$\frac{\sin (20.64381859)}{a} = \frac{\sin (32)}{100}$

Step 19

Solve the equation for $a$ .

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Step 19.1

Factor each term.

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Step 19.1.1

Evaluate $\sin (20.64381859)$ .

$\frac{0.35255742}{a} = \frac{\sin (32)}{100}$

Step 19.1.2

Evaluate $\sin (32)$ .

$\frac{0.35255742}{a} = \frac{0.52991926}{100}$

Step 19.1.3

Divide $0.52991926$ by $100$ .

$\frac{0.35255742}{a} = 0.00529919$

Step 19.2

Find the LCD of the terms in the equation.

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Step 19.2.1

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

$a, 1$

Step 19.2.2

The LCM of one and any expression is the expression.

$a$

Step 19.3

Multiply each term in $\frac{0.35255742}{a} = 0.00529919$ by $a$ to eliminate the fractions.

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Step 19.3.1

Multiply each term in $\frac{0.35255742}{a} = 0.00529919$ by $a$ .

$\frac{0.35255742}{a} a = 0.00529919 a$

Step 19.3.2

Simplify the left side.

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Step 19.3.2.1

Cancel the common factor of $a$ .

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Step 19.3.2.1.1

Cancel the common factor.

$\frac{0.35255742}{a} a = 0.00529919 a$

Step 19.3.2.1.2

Rewrite the expression.

$0.35255742 = 0.00529919 a$

Step 19.4

Solve the equation.

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Step 19.4.1

Rewrite the equation as $0.00529919 a = 0.35255742$ .

$0.00529919 a = 0.35255742$

Step 19.4.2

Divide each term in $0.00529919 a = 0.35255742$ by $0.00529919$ and simplify.

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Step 19.4.2.1

Divide each term in $0.00529919 a = 0.35255742$ by $0.00529919$ .

$\frac{0.00529919 a}{0.00529919} = \frac{0.35255742}{0.00529919}$

Step 19.4.2.2

Simplify the left side.

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Step 19.4.2.2.1

Cancel the common factor of $0.00529919$ .

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Step 19.4.2.2.1.1

Cancel the common factor.

$\frac{0.00529919 a}{0.00529919} = \frac{0.35255742}{0.00529919}$

Step 19.4.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{0.35255742}{0.00529919}$

Step 19.4.2.3

Simplify the right side.

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Step 19.4.2.3.1

Divide $0.35255742$ by $0.00529919$ .

$a = 66.53040335$

Step 20

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

First Triangle Combination:

$A = 95.3561814$

$B = 32$

$C = 52.64381859$

$a = 50 \sqrt{13 - 12 \cos (95.3561814)}$

$b = 100$

$c = 150$

Second Triangle Combination:

$A = 20.64381859$

$B = 32$

$C = 127.3561814$

$a = 66.53040335$

$b = 100$

$c = 150$