Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 12 \\ c = \\ a = & 10 \end{matrix} & \begin{matrix} A = & 48 \\ B = \\ C = \end{matrix} \end{matrix}$

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 $B$ .

$\frac{\sin (B)}{12} = \frac{\sin (48)}{10}$

Step 4

Solve the equation for $B$ .

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Step 4.1

Multiply both sides of the equation by $12$ .

$12 \frac{\sin (B)}{12} = 12 \frac{\sin (48)}{10}$

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 $12$ .

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Step 4.2.1.1.1

Cancel the common factor.

$12 \frac{\sin (B)}{12} = 12 \frac{\sin (48)}{10}$

Step 4.2.1.1.2

Rewrite the expression.

$\sin (B) = 12 \frac{\sin (48)}{10}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $12 \frac{\sin (48)}{10}$ .

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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 $12$ .

$\sin (B) = 2 (6) \frac{\sin (48)}{10}$

Step 4.2.2.1.1.2

Factor $2$ out of $10$ .

$\sin (B) = 2 \cdot 6 \frac{\sin (48)}{2 \cdot 5}$

Step 4.2.2.1.1.3

Cancel the common factor.

$\sin (B) = 2 \cdot 6 \frac{\sin (48)}{2 \cdot 5}$

Step 4.2.2.1.1.4

Rewrite the expression.

$\sin (B) = 6 \frac{\sin (48)}{5}$

Step 4.2.2.1.2

Combine $6$ and $\frac{\sin (48)}{5}$ .

$\sin (B) = \frac{6 \sin (48)}{5}$

Step 4.2.2.1.3

Evaluate $\sin (48)$ .

$\sin (B) = \frac{6 \cdot 0.74314482}{5}$

Step 4.2.2.1.4

Multiply $6$ by $0.74314482$ .

$\sin (B) = \frac{4.45886895}{5}$

Step 4.2.2.1.5

Divide $4.45886895$ by $5$ .

$\sin (B) = 0.89177379$

Step 4.3

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

$B = \arcsin (0.89177379)$

Step 4.4

Simplify the right side.

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Step 4.4.1

Evaluate $\arcsin (0.89177379)$ .

$B = 63.09699387$

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.

$B = 180 - 63.09699387$

Step 4.6

Subtract $63.09699387$ from $180$ .

$B = 116.90300612$

Step 4.7

The solution to the equation $B = 63.09699387$ .

$B = 63.09699387, 116.90300612$

Step 5

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

$48 + C + 63.09699387 = 180$

Step 6

Solve the equation for $C$ .

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Step 6.1

Add $48$ and $63.09699387$ .

$C + 111.09699387 = 180$

Step 6.2

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

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Step 6.2.1

Subtract $111.09699387$ from both sides of the equation.

$C = 180 - 111.09699387$

Step 6.2.2

Subtract $111.09699387$ from $180$ .

$C = 68.90300612$

Step 7

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

Step 8

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

$\frac{\sin (68.90300612)}{c} = \frac{\sin (48)}{10}$

Step 9

Solve the equation for $c$ .

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Step 9.1

Factor each term.

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Step 9.1.1

Evaluate $\sin (68.90300612)$ .

$\frac{0.93297242}{c} = \frac{\sin (48)}{10}$

Step 9.1.2

Evaluate $\sin (48)$ .

$\frac{0.93297242}{c} = \frac{0.74314482}{10}$

Step 9.1.3

Divide $0.74314482$ by $10$ .

$\frac{0.93297242}{c} = 0.07431448$

Step 9.2

Find the LCD of the terms in the equation.

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Step 9.2.1

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

$c, 1$

Step 9.2.2

The LCM of one and any expression is the expression.

$c$

Step 9.3

Multiply each term in $\frac{0.93297242}{c} = 0.07431448$ by $c$ to eliminate the fractions.

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Step 9.3.1

Multiply each term in $\frac{0.93297242}{c} = 0.07431448$ by $c$ .

$\frac{0.93297242}{c} c = 0.07431448 c$

Step 9.3.2

Simplify the left side.

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Step 9.3.2.1

Cancel the common factor of $c$ .

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Step 9.3.2.1.1

Cancel the common factor.

$\frac{0.93297242}{c} c = 0.07431448 c$

Step 9.3.2.1.2

Rewrite the expression.

$0.93297242 = 0.07431448 c$

Step 9.4

Solve the equation.

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Step 9.4.1

Rewrite the equation as $0.07431448 c = 0.93297242$ .

$0.07431448 c = 0.93297242$

Step 9.4.2

Divide each term in $0.07431448 c = 0.93297242$ by $0.07431448$ and simplify.

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Step 9.4.2.1

Divide each term in $0.07431448 c = 0.93297242$ by $0.07431448$ .

$\frac{0.07431448 c}{0.07431448} = \frac{0.93297242}{0.07431448}$

Step 9.4.2.2

Simplify the left side.

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Step 9.4.2.2.1

Cancel the common factor of $0.07431448$ .

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Step 9.4.2.2.1.1

Cancel the common factor.

$\frac{0.07431448 c}{0.07431448} = \frac{0.93297242}{0.07431448}$

Step 9.4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{0.93297242}{0.07431448}$

Step 9.4.2.3

Simplify the right side.

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Step 9.4.2.3.1

Divide $0.93297242$ by $0.07431448$ .

$c = 12.55438226$

Step 10

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

Solve for the second triangle.

Step 11

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

Step 12

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

$\frac{\sin (B)}{12} = \frac{\sin (48)}{10}$

Step 13

Solve the equation for $B$ .

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Step 13.1

Multiply both sides of the equation by $12$ .

$12 \frac{\sin (B)}{12} = 12 \frac{\sin (48)}{10}$

Step 13.2

Simplify both sides of the equation.

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Step 13.2.1

Simplify the left side.

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Step 13.2.1.1

Cancel the common factor of $12$ .

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Step 13.2.1.1.1

Cancel the common factor.

$12 \frac{\sin (B)}{12} = 12 \frac{\sin (48)}{10}$

Step 13.2.1.1.2

Rewrite the expression.

$\sin (B) = 12 \frac{\sin (48)}{10}$

Step 13.2.2

Simplify the right side.

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Step 13.2.2.1

Simplify $12 \frac{\sin (48)}{10}$ .

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Step 13.2.2.1.1

Cancel the common factor of $2$ .

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Step 13.2.2.1.1.1

Factor $2$ out of $12$ .

$\sin (B) = 2 (6) \frac{\sin (48)}{10}$

Step 13.2.2.1.1.2

Factor $2$ out of $10$ .

$\sin (B) = 2 \cdot 6 \frac{\sin (48)}{2 \cdot 5}$

Step 13.2.2.1.1.3

Cancel the common factor.

$\sin (B) = 2 \cdot 6 \frac{\sin (48)}{2 \cdot 5}$

Step 13.2.2.1.1.4

Rewrite the expression.

$\sin (B) = 6 \frac{\sin (48)}{5}$

Step 13.2.2.1.2

Combine $6$ and $\frac{\sin (48)}{5}$ .

$\sin (B) = \frac{6 \sin (48)}{5}$

Step 13.2.2.1.3

Evaluate $\sin (48)$ .

$\sin (B) = \frac{6 \cdot 0.74314482}{5}$

Step 13.2.2.1.4

Multiply $6$ by $0.74314482$ .

$\sin (B) = \frac{4.45886895}{5}$

Step 13.2.2.1.5

Divide $4.45886895$ by $5$ .

$\sin (B) = 0.89177379$

Step 13.3

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

$B = \arcsin (0.89177379)$

Step 13.4

Simplify the right side.

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Step 13.4.1

Evaluate $\arcsin (0.89177379)$ .

$B = 63.09699387$

Step 13.5

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.

$B = 180 - 63.09699387$

Step 13.6

Subtract $63.09699387$ from $180$ .

$B = 116.90300612$

Step 13.7

The solution to the equation $B = 63.09699387$ .

$B = 63.09699387, 116.90300612$

Step 14

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

$48 + C + 116.90300612 = 180$

Step 15

Solve the equation for $C$ .

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Step 15.1

Add $48$ and $116.90300612$ .

$C + 164.90300612 = 180$

Step 15.2

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

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Step 15.2.1

Subtract $164.90300612$ from both sides of the equation.

$C = 180 - 164.90300612$

Step 15.2.2

Subtract $164.90300612$ from $180$ .

$C = 15.09699387$

Step 16

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

Step 17

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

$\frac{\sin (15.09699387)}{c} = \frac{\sin (48)}{10}$

Step 18

Solve the equation for $c$ .

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Step 18.1

Factor each term.

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Step 18.1.1

Evaluate $\sin (15.09699387)$ .

$\frac{0.26045385}{c} = \frac{\sin (48)}{10}$

Step 18.1.2

Evaluate $\sin (48)$ .

$\frac{0.26045385}{c} = \frac{0.74314482}{10}$

Step 18.1.3

Divide $0.74314482$ by $10$ .

$\frac{0.26045385}{c} = 0.07431448$

Step 18.2

Find the LCD of the terms in the equation.

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Step 18.2.1

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

$c, 1$

Step 18.2.2

The LCM of one and any expression is the expression.

$c$

Step 18.3

Multiply each term in $\frac{0.26045385}{c} = 0.07431448$ by $c$ to eliminate the fractions.

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Step 18.3.1

Multiply each term in $\frac{0.26045385}{c} = 0.07431448$ by $c$ .

$\frac{0.26045385}{c} c = 0.07431448 c$

Step 18.3.2

Simplify the left side.

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Step 18.3.2.1

Cancel the common factor of $c$ .

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Step 18.3.2.1.1

Cancel the common factor.

$\frac{0.26045385}{c} c = 0.07431448 c$

Step 18.3.2.1.2

Rewrite the expression.

$0.26045385 = 0.07431448 c$

Step 18.4

Solve the equation.

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Step 18.4.1

Rewrite the equation as $0.07431448 c = 0.26045385$ .

$0.07431448 c = 0.26045385$

Step 18.4.2

Divide each term in $0.07431448 c = 0.26045385$ by $0.07431448$ and simplify.

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Step 18.4.2.1

Divide each term in $0.07431448 c = 0.26045385$ by $0.07431448$ .

$\frac{0.07431448 c}{0.07431448} = \frac{0.26045385}{0.07431448}$

Step 18.4.2.2

Simplify the left side.

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Step 18.4.2.2.1

Cancel the common factor of $0.07431448$ .

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Step 18.4.2.2.1.1

Cancel the common factor.

$\frac{0.07431448 c}{0.07431448} = \frac{0.26045385}{0.07431448}$

Step 18.4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{0.26045385}{0.07431448}$

Step 18.4.2.3

Simplify the right side.

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Step 18.4.2.3.1

Divide $0.26045385$ by $0.07431448$ .

$c = 3.50475229$

Step 19

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

First Triangle Combination:

$A = 48$

$B = 63.09699387$

$C = 68.90300612$

$a = 10$

$b = 12$

$c = 12.55438226$

Second Triangle Combination:

$A = 48$

$B = 116.90300612$

$C = 15.09699387$

$a = 10$

$b = 12$

$c = 3.50475229$