Trigonometry Examples

$A = 32$ , $a = 19$ , $b = 14$

Step 1

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 2

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

$\frac{\sin (B)}{14} = \frac{\sin (32)}{19}$

Step 3

Solve the equation for $B$ .

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Step 3.1

Multiply both sides of the equation by $14$ .

$14 \frac{\sin (B)}{14} = 14 \frac{\sin (32)}{19}$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Cancel the common factor of $14$ .

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Step 3.2.1.1.1

Cancel the common factor.

$14 \frac{\sin (B)}{14} = 14 \frac{\sin (32)}{19}$

Step 3.2.1.1.2

Rewrite the expression.

$\sin (B) = 14 \frac{\sin (32)}{19}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $14 \frac{\sin (32)}{19}$ .

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Step 3.2.2.1.1

Evaluate $\sin (32)$ .

$\sin (B) = 14 (\frac{0.52991926}{19})$

Step 3.2.2.1.2

Divide $0.52991926$ by $19$ .

$\sin (B) = 14 \cdot 0.02789048$

Step 3.2.2.1.3

Multiply $14$ by $0.02789048$ .

$\sin (B) = 0.39046682$

Step 3.3

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

$B = \arcsin (0.39046682)$

Step 3.4

Simplify the right side.

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Step 3.4.1

Evaluate $\arcsin (0.39046682)$ .

$B = 22.98354981$

Step 3.5

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.

$B = 180 - 22.98354981$

Step 3.6

Subtract $22.98354981$ from $180$ .

$B = 157.01645018$

Step 3.7

The solution to the equation $B = 22.98354981$ .

$B = 22.98354981, 157.01645018$

Step 3.8

Exclude the invalid angle.

$B = 22.98354981$

Step 4

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

$32 + C + 22.98354981 = 180$

Step 5

Solve the equation for $C$ .

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Step 5.1

Add $32$ and $22.98354981$ .

$C + 54.98354981 = 180$

Step 5.2

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

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Step 5.2.1

Subtract $54.98354981$ from both sides of the equation.

$C = 180 - 54.98354981$

Step 5.2.2

Subtract $54.98354981$ from $180$ .

$C = 125.01645018$

Step 6

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

Step 7

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

$\frac{\sin (125.01645018)}{c} = \frac{\sin (32)}{19}$

Step 8

Solve the equation for $c$ .

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Step 8.1

Factor each term.

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Step 8.1.1

Evaluate $\sin (125.01645018)$ .

$\frac{0.81898733}{c} = \frac{\sin (32)}{19}$

Step 8.1.2

Evaluate $\sin (32)$ .

$\frac{0.81898733}{c} = \frac{0.52991926}{19}$

Step 8.1.3

Divide $0.52991926$ by $19$ .

$\frac{0.81898733}{c} = 0.02789048$

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.

$c, 1$

Step 8.2.2

The LCM of one and any expression is the expression.

$c$

Step 8.3

Multiply each term in $\frac{0.81898733}{c} = 0.02789048$ by $c$ to eliminate the fractions.

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Step 8.3.1

Multiply each term in $\frac{0.81898733}{c} = 0.02789048$ by $c$ .

$\frac{0.81898733}{c} c = 0.02789048 c$

Step 8.3.2

Simplify the left side.

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Step 8.3.2.1

Cancel the common factor of $c$ .

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Step 8.3.2.1.1

Cancel the common factor.

$\frac{0.81898733}{c} c = 0.02789048 c$

Step 8.3.2.1.2

Rewrite the expression.

$0.81898733 = 0.02789048 c$

Step 8.4

Solve the equation.

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Step 8.4.1

Rewrite the equation as $0.02789048 c = 0.81898733$ .

$0.02789048 c = 0.81898733$

Step 8.4.2

Divide each term in $0.02789048 c = 0.81898733$ by $0.02789048$ and simplify.

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Step 8.4.2.1

Divide each term in $0.02789048 c = 0.81898733$ by $0.02789048$ .

$\frac{0.02789048 c}{0.02789048} = \frac{0.81898733}{0.02789048}$

Step 8.4.2.2

Simplify the left side.

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Step 8.4.2.2.1

Cancel the common factor of $0.02789048$ .

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Step 8.4.2.2.1.1

Cancel the common factor.

$\frac{0.02789048 c}{0.02789048} = \frac{0.81898733}{0.02789048}$

Step 8.4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{0.81898733}{0.02789048}$

Step 8.4.2.3

Simplify the right side.

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Step 8.4.2.3.1

Divide $0.81898733$ by $0.02789048$ .

$c = 29.36439631$

Step 9

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

$A = 32$

$B = 22.98354981$

$C = 125.01645018$

$a = 19$

$b = 14$

$c = 29.36439631$