Trigonometry Examples

$a = 2$ , $b = 3$ , $A = 20$

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 $B$ .

$\frac{\sin (B)}{3} = \frac{\sin (20)}{2}$

Step 4

Solve the equation for $B$ .

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Step 4.1

Multiply both sides of the equation by $3$ .

$3 \frac{\sin (B)}{3} = 3 \frac{\sin (20)}{2}$

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 $3$ .

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Step 4.2.1.1.1

Cancel the common factor.

$3 \frac{\sin (B)}{3} = 3 \frac{\sin (20)}{2}$

Step 4.2.1.1.2

Rewrite the expression.

$\sin (B) = 3 \frac{\sin (20)}{2}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $3 \frac{\sin (20)}{2}$ .

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Step 4.2.2.1.1

Evaluate $\sin (20)$ .

$\sin (B) = 3 (\frac{0.34202014}{2})$

Step 4.2.2.1.2

Divide $0.34202014$ by $2$ .

$\sin (B) = 3 \cdot 0.17101007$

Step 4.2.2.1.3

Multiply $3$ by $0.17101007$ .

$\sin (B) = 0.51303021$

Step 4.3

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

$B = \arcsin (0.51303021)$

Step 4.4

Simplify the right side.

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Step 4.4.1

Evaluate $\arcsin (0.51303021)$ .

$B = 30.86588247$

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.

$B = 180 - 30.86588247$

Step 4.6

Subtract $30.86588247$ from $180$ .

$B = 149.13411752$

Step 4.7

The solution to the equation $B = 30.86588247$ .

$B = 30.86588247, 149.13411752$

Step 5

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

$20 + C + 30.86588247 = 180$

Step 6

Solve the equation for $C$ .

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Step 6.1

Add $20$ and $30.86588247$ .

$C + 50.86588247 = 180$

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 $50.86588247$ from both sides of the equation.

$C = 180 - 50.86588247$

Step 6.2.2

Subtract $50.86588247$ from $180$ .

$C = 129.13411752$

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 $c$ .

$\frac{\sin (129.13411752)}{c} = \frac{\sin (20)}{2}$

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 $\sin (129.13411752)$ .

$\frac{0.77567072}{c} = \frac{\sin (20)}{2}$

Step 9.1.2

Evaluate $\sin (20)$ .

$\frac{0.77567072}{c} = \frac{0.34202014}{2}$

Step 9.1.3

Divide $0.34202014$ by $2$ .

$\frac{0.77567072}{c} = 0.17101007$

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 $\frac{0.77567072}{c} = 0.17101007$ by $c$ to eliminate the fractions.

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Step 9.3.1

Multiply each term in $\frac{0.77567072}{c} = 0.17101007$ by $c$ .

$\frac{0.77567072}{c} c = 0.17101007 c$

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.

$\frac{0.77567072}{c} c = 0.17101007 c$

Step 9.3.2.1.2

Rewrite the expression.

$0.77567072 = 0.17101007 c$

Step 9.4

Solve the equation.

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Step 9.4.1

Rewrite the equation as $0.17101007 c = 0.77567072$ .

$0.17101007 c = 0.77567072$

Step 9.4.2

Divide each term in $0.17101007 c = 0.77567072$ by $0.17101007$ and simplify.

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Step 9.4.2.1

Divide each term in $0.17101007 c = 0.77567072$ by $0.17101007$ .

$\frac{0.17101007 c}{0.17101007} = \frac{0.77567072}{0.17101007}$

Step 9.4.2.2

Simplify the left side.

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Step 9.4.2.2.1

Cancel the common factor of $0.17101007$ .

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Step 9.4.2.2.1.1

Cancel the common factor.

$\frac{0.17101007 c}{0.17101007} = \frac{0.77567072}{0.17101007}$

Step 9.4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{0.77567072}{0.17101007}$

Step 9.4.2.3

Simplify the right side.

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Step 9.4.2.3.1

Divide $0.77567072$ by $0.17101007$ .

$c = 4.53581895$

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 $B$ .

$\frac{\sin (B)}{3} = \frac{\sin (20)}{2}$

Step 13

Solve the equation for $B$ .

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Step 13.1

Multiply both sides of the equation by $3$ .

$3 \frac{\sin (B)}{3} = 3 \frac{\sin (20)}{2}$

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 $3$ .

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Step 13.2.1.1.1

Cancel the common factor.

$3 \frac{\sin (B)}{3} = 3 \frac{\sin (20)}{2}$

Step 13.2.1.1.2

Rewrite the expression.

$\sin (B) = 3 \frac{\sin (20)}{2}$

Step 13.2.2

Simplify the right side.

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Step 13.2.2.1

Simplify $3 \frac{\sin (20)}{2}$ .

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Step 13.2.2.1.1

Evaluate $\sin (20)$ .

$\sin (B) = 3 (\frac{0.34202014}{2})$

Step 13.2.2.1.2

Divide $0.34202014$ by $2$ .

$\sin (B) = 3 \cdot 0.17101007$

Step 13.2.2.1.3

Multiply $3$ by $0.17101007$ .

$\sin (B) = 0.51303021$

Step 13.3

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

$B = \arcsin (0.51303021)$

Step 13.4

Simplify the right side.

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Step 13.4.1

Evaluate $\arcsin (0.51303021)$ .

$B = 30.86588247$

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.

$B = 180 - 30.86588247$

Step 13.6

Subtract $30.86588247$ from $180$ .

$B = 149.13411752$

Step 13.7

The solution to the equation $B = 30.86588247$ .

$B = 30.86588247, 149.13411752$

Step 14

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

$20 + C + 149.13411752 = 180$

Step 15

Solve the equation for $C$ .

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Step 15.1

Add $20$ and $149.13411752$ .

$C + 169.13411752 = 180$

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 $169.13411752$ from both sides of the equation.

$C = 180 - 169.13411752$

Step 15.2.2

Subtract $169.13411752$ from $180$ .

$C = 10.86588247$

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 $c$ .

$\frac{\sin (10.86588247)}{c} = \frac{\sin (20)}{2}$

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 $\sin (10.86588247)$ .

$\frac{0.18851068}{c} = \frac{\sin (20)}{2}$

Step 18.1.2

Evaluate $\sin (20)$ .

$\frac{0.18851068}{c} = \frac{0.34202014}{2}$

Step 18.1.3

Divide $0.34202014$ by $2$ .

$\frac{0.18851068}{c} = 0.17101007$

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 $\frac{0.18851068}{c} = 0.17101007$ by $c$ to eliminate the fractions.

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Step 18.3.1

Multiply each term in $\frac{0.18851068}{c} = 0.17101007$ by $c$ .

$\frac{0.18851068}{c} c = 0.17101007 c$

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.

$\frac{0.18851068}{c} c = 0.17101007 c$

Step 18.3.2.1.2

Rewrite the expression.

$0.18851068 = 0.17101007 c$

Step 18.4

Solve the equation.

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Step 18.4.1

Rewrite the equation as $0.17101007 c = 0.18851068$ .

$0.17101007 c = 0.18851068$

Step 18.4.2

Divide each term in $0.17101007 c = 0.18851068$ by $0.17101007$ and simplify.

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Step 18.4.2.1

Divide each term in $0.17101007 c = 0.18851068$ by $0.17101007$ .

$\frac{0.17101007 c}{0.17101007} = \frac{0.18851068}{0.17101007}$

Step 18.4.2.2

Simplify the left side.

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Step 18.4.2.2.1

Cancel the common factor of $0.17101007$ .

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Step 18.4.2.2.1.1

Cancel the common factor.

$\frac{0.17101007 c}{0.17101007} = \frac{0.18851068}{0.17101007}$

Step 18.4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{0.18851068}{0.17101007}$

Step 18.4.2.3

Simplify the right side.

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Step 18.4.2.3.1

Divide $0.18851068$ by $0.17101007$ .

$c = 1.10233676$

Step 19

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

First Triangle Combination:

$A = 20$

$B = 30.86588247$

$C = 129.13411752$

$a = 2$

$b = 3$

$c = 4.53581895$

Second Triangle Combination:

$A = 20$

$B = 149.13411752$

$C = 10.86588247$

$a = 2$

$b = 3$

$c = 1.10233676$