Trigonometry Examples

$b = 118$ , $c = 36$ , $C = 14$

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 $B$ .

$\frac{\sin (B)}{118} = \frac{\sin (14)}{36}$

Step 4

Solve the equation for $B$ .

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Step 4.1

Multiply both sides of the equation by $118$ .

$118 \frac{\sin (B)}{118} = 118 \frac{\sin (14)}{36}$

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 $118$ .

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Step 4.2.1.1.1

Cancel the common factor.

$118 \frac{\sin (B)}{118} = 118 \frac{\sin (14)}{36}$

Step 4.2.1.1.2

Rewrite the expression.

$\sin (B) = 118 \frac{\sin (14)}{36}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $118 \frac{\sin (14)}{36}$ .

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Step 4.2.2.1.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.1

Factor $2$ out of $118$ .

$\sin (B) = 2 (59) \frac{\sin (14)}{36}$

Step 4.2.2.1.1.2

Factor $2$ out of $36$ .

$\sin (B) = 2 \cdot 59 \frac{\sin (14)}{2 \cdot 18}$

Step 4.2.2.1.1.3

Cancel the common factor.

$\sin (B) = 2 \cdot 59 \frac{\sin (14)}{2 \cdot 18}$

Step 4.2.2.1.1.4

Rewrite the expression.

$\sin (B) = 59 \frac{\sin (14)}{18}$

Step 4.2.2.1.2

Combine $59$ and $\frac{\sin (14)}{18}$ .

$\sin (B) = \frac{59 \sin (14)}{18}$

Step 4.2.2.1.3

Evaluate $\sin (14)$ .

$\sin (B) = \frac{59 \cdot 0.24192189}{18}$

Step 4.2.2.1.4

Multiply $59$ by $0.24192189$ .

$\sin (B) = \frac{14.27339184}{18}$

Step 4.2.2.1.5

Divide $14.27339184$ by $18$ .

$\sin (B) = 0.79296621$

Step 4.3

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

$B = \arcsin (0.79296621)$

Step 4.4

Simplify the right side.

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Step 4.4.1

Evaluate $\arcsin (0.79296621)$ .

$B = 52.46357923$

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.

$B = 180 - 52.46357923$

Step 4.6

Subtract $52.46357923$ from $180$ .

$B = 127.53642076$

Step 4.7

The solution to the equation $B = 52.46357923$ .

$B = 52.46357923, 127.53642076$

Step 5

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

$A + 14 + 52.46357923 = 180$

Step 6

Solve the equation for $A$ .

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Step 6.1

Add $14$ and $52.46357923$ .

$A + 66.46357923 = 180$

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 $66.46357923$ from both sides of the equation.

$A = 180 - 66.46357923$

Step 6.2.2

Subtract $66.46357923$ from $180$ .

$A = 113.53642076$

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.

$a = \sqrt{{(118)}^{2} + {(36)}^{2} - 2 \cdot 118 \cdot 36 \cos (113.53642076)}$

Step 10

Simplify the results.

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Step 10.1

Raise $118$ to the power of $2$ .

$a = \sqrt{13924 + {(36)}^{2} - 2 \cdot 118 \cdot (36 \cos (113.53642076))}$

Step 10.2

Raise $36$ to the power of $2$ .

$a = \sqrt{13924 + 1296 - 2 \cdot 118 \cdot (36 \cos (113.53642076))}$

Step 10.3

Multiply $- 2 \cdot 118 \cdot 36$ .

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Step 10.3.1

Multiply $- 2$ by $118$ .

$a = \sqrt{13924 + 1296 - 236 \cdot (36 \cos (113.53642076))}$

Step 10.3.2

Multiply $- 236$ by $36$ .

$a = \sqrt{13924 + 1296 - 8496 \cos (113.53642076)}$

Step 10.4

Add $13924$ and $1296$ .

$a = \sqrt{15220 - 8496 \cos (113.53642076)}$

Step 10.5

Rewrite $15220 - 8496 \cos (113.53642076)$ as $2^{2} (3805 - 2124 \cos (113.53642076))$ .

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Step 10.5.1

Factor $4$ out of $15220$ .

$a = \sqrt{4 (3805) - 8496 \cos (113.53642076)}$

Step 10.5.2

Factor $4$ out of $- 8496 \cos (113.53642076)$ .

$a = \sqrt{4 (3805) + 4 (- 2124 \cos (113.53642076))}$

Step 10.5.3

Factor $4$ out of $4 (3805) + 4 (- 2124 \cos (113.53642076))$ .

$a = \sqrt{4 (3805 - 2124 \cos (113.53642076))}$

Step 10.5.4

Rewrite $4$ as $2^{2}$ .

$a = \sqrt{2^{2} (3805 - 2124 \cos (113.53642076))}$

Step 10.6

Pull terms out from under the radical.

$a = 2 \sqrt{3805 - 2124 \cos (113.53642076)}$

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 $B$ .

$\frac{\sin (B)}{118} = \frac{\sin (14)}{36}$

Step 14

Solve the equation for $B$ .

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Step 14.1

Multiply both sides of the equation by $118$ .

$118 \frac{\sin (B)}{118} = 118 \frac{\sin (14)}{36}$

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 $118$ .

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Step 14.2.1.1.1

Cancel the common factor.

$118 \frac{\sin (B)}{118} = 118 \frac{\sin (14)}{36}$

Step 14.2.1.1.2

Rewrite the expression.

$\sin (B) = 118 \frac{\sin (14)}{36}$

Step 14.2.2

Simplify the right side.

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Step 14.2.2.1

Simplify $118 \frac{\sin (14)}{36}$ .

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Step 14.2.2.1.1

Cancel the common factor of $2$ .

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Step 14.2.2.1.1.1

Factor $2$ out of $118$ .

$\sin (B) = 2 (59) \frac{\sin (14)}{36}$

Step 14.2.2.1.1.2

Factor $2$ out of $36$ .

$\sin (B) = 2 \cdot 59 \frac{\sin (14)}{2 \cdot 18}$

Step 14.2.2.1.1.3

Cancel the common factor.

$\sin (B) = 2 \cdot 59 \frac{\sin (14)}{2 \cdot 18}$

Step 14.2.2.1.1.4

Rewrite the expression.

$\sin (B) = 59 \frac{\sin (14)}{18}$

Step 14.2.2.1.2

Combine $59$ and $\frac{\sin (14)}{18}$ .

$\sin (B) = \frac{59 \sin (14)}{18}$

Step 14.2.2.1.3

Evaluate $\sin (14)$ .

$\sin (B) = \frac{59 \cdot 0.24192189}{18}$

Step 14.2.2.1.4

Multiply $59$ by $0.24192189$ .

$\sin (B) = \frac{14.27339184}{18}$

Step 14.2.2.1.5

Divide $14.27339184$ by $18$ .

$\sin (B) = 0.79296621$

Step 14.3

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

$B = \arcsin (0.79296621)$

Step 14.4

Simplify the right side.

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Step 14.4.1

Evaluate $\arcsin (0.79296621)$ .

$B = 52.46357923$

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.

$B = 180 - 52.46357923$

Step 14.6

Subtract $52.46357923$ from $180$ .

$B = 127.53642076$

Step 14.7

The solution to the equation $B = 52.46357923$ .

$B = 52.46357923, 127.53642076$

Step 15

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

$A + 14 + 127.53642076 = 180$

Step 16

Solve the equation for $A$ .

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Step 16.1

Add $14$ and $127.53642076$ .

$A + 141.53642076 = 180$

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 $141.53642076$ from both sides of the equation.

$A = 180 - 141.53642076$

Step 16.2.2

Subtract $141.53642076$ from $180$ .

$A = 38.46357923$

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 $a$ .

$\frac{\sin (38.46357923)}{a} = \frac{\sin (127.53642076)}{118}$

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 $\sin (38.46357923)$ .

$\frac{0.62201703}{a} = \frac{\sin (127.53642076)}{118}$

Step 19.1.2

Evaluate $\sin (127.53642076)$ .

$\frac{0.62201703}{a} = \frac{0.79296621}{118}$

Step 19.1.3

Divide $0.79296621$ by $118$ .

$\frac{0.62201703}{a} = 0.00672005$

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 $\frac{0.62201703}{a} = 0.00672005$ by $a$ to eliminate the fractions.

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Step 19.3.1

Multiply each term in $\frac{0.62201703}{a} = 0.00672005$ by $a$ .

$\frac{0.62201703}{a} a = 0.00672005 a$

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.

$\frac{0.62201703}{a} a = 0.00672005 a$

Step 19.3.2.1.2

Rewrite the expression.

$0.62201703 = 0.00672005 a$

Step 19.4

Solve the equation.

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Step 19.4.1

Rewrite the equation as $0.00672005 a = 0.62201703$ .

$0.00672005 a = 0.62201703$

Step 19.4.2

Divide each term in $0.00672005 a = 0.62201703$ by $0.00672005$ and simplify.

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Step 19.4.2.1

Divide each term in $0.00672005 a = 0.62201703$ by $0.00672005$ .

$\frac{0.00672005 a}{0.00672005} = \frac{0.62201703}{0.00672005}$

Step 19.4.2.2

Simplify the left side.

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Step 19.4.2.2.1

Cancel the common factor of $0.00672005$ .

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Step 19.4.2.2.1.1

Cancel the common factor.

$\frac{0.00672005 a}{0.00672005} = \frac{0.62201703}{0.00672005}$

Step 19.4.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{0.62201703}{0.00672005}$

Step 19.4.2.3

Simplify the right side.

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Step 19.4.2.3.1

Divide $0.62201703$ by $0.00672005$ .

$a = 92.56133371$

Step 20

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

First Triangle Combination:

$A = 113.53642076$

$B = 52.46357923$

$C = 14$

$a = 2 \sqrt{3805 - 2124 \cos (113.53642076)}$

$b = 118$

$c = 36$

Second Triangle Combination:

$A = 38.46357923$

$B = 127.53642076$

$C = 14$

$a = 92.56133371$

$b = 118$

$c = 36$