Trigonometry Examples

$a = 14$ , $b = 8$ , $A = 16$

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 $B$ .

$\frac{\sin (B)}{8} = \frac{\sin (16)}{14}$

Step 3

Solve the equation for $B$ .

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Step 3.1

Multiply both sides of the equation by $8$ .

$8 \frac{\sin (B)}{8} = 8 \frac{\sin (16)}{14}$

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 $8$ .

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Step 3.2.1.1.1

Cancel the common factor.

$8 \frac{\sin (B)}{8} = 8 \frac{\sin (16)}{14}$

Step 3.2.1.1.2

Rewrite the expression.

$\sin (B) = 8 \frac{\sin (16)}{14}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $8 \frac{\sin (16)}{14}$ .

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Step 3.2.2.1.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1.1

Factor $2$ out of $8$ .

$\sin (B) = 2 (4) \frac{\sin (16)}{14}$

Step 3.2.2.1.1.2

Factor $2$ out of $14$ .

$\sin (B) = 2 \cdot 4 \frac{\sin (16)}{2 \cdot 7}$

Step 3.2.2.1.1.3

Cancel the common factor.

$\sin (B) = 2 \cdot 4 \frac{\sin (16)}{2 \cdot 7}$

Step 3.2.2.1.1.4

Rewrite the expression.

$\sin (B) = 4 \frac{\sin (16)}{7}$

Step 3.2.2.1.2

Combine $4$ and $\frac{\sin (16)}{7}$ .

$\sin (B) = \frac{4 \sin (16)}{7}$

Step 3.2.2.1.3

Evaluate $\sin (16)$ .

$\sin (B) = \frac{4 \cdot 0.27563735}{7}$

Step 3.2.2.1.4

Multiply $4$ by $0.27563735$ .

$\sin (B) = \frac{1.10254942}{7}$

Step 3.2.2.1.5

Divide $1.10254942$ by $7$ .

$\sin (B) = 0.15750706$

Step 3.3

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

$B = \arcsin (0.15750706)$

Step 3.4

Simplify the right side.

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Step 3.4.1

Evaluate $\arcsin (0.15750706)$ .

$B = 9.0622266$

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.

$B = 180 - 9.0622266$

Step 3.6

Subtract $9.0622266$ from $180$ .

$B = 170.93777339$

Step 3.7

The solution to the equation $B = 9.0622266$ .

$B = 9.0622266, 170.93777339$

Step 3.8

Exclude the invalid angle.

$B = 9.0622266$

Step 4

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

$16 + C + 9.0622266 = 180$

Step 5

Solve the equation for $C$ .

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Step 5.1

Add $16$ and $9.0622266$ .

$C + 25.0622266 = 180$

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 $25.0622266$ from both sides of the equation.

$C = 180 - 25.0622266$

Step 5.2.2

Subtract $25.0622266$ from $180$ .

$C = 154.93777339$

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 $c$ .

$\frac{\sin (154.93777339)}{c} = \frac{\sin (16)}{14}$

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 $\sin (154.93777339)$ .

$\frac{0.42360231}{c} = \frac{\sin (16)}{14}$

Step 8.1.2

Evaluate $\sin (16)$ .

$\frac{0.42360231}{c} = \frac{0.27563735}{14}$

Step 8.1.3

Divide $0.27563735$ by $14$ .

$\frac{0.42360231}{c} = 0.01968838$

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 $\frac{0.42360231}{c} = 0.01968838$ by $c$ to eliminate the fractions.

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Step 8.3.1

Multiply each term in $\frac{0.42360231}{c} = 0.01968838$ by $c$ .

$\frac{0.42360231}{c} c = 0.01968838 c$

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.

$\frac{0.42360231}{c} c = 0.01968838 c$

Step 8.3.2.1.2

Rewrite the expression.

$0.42360231 = 0.01968838 c$

Step 8.4

Solve the equation.

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Step 8.4.1

Rewrite the equation as $0.01968838 c = 0.42360231$ .

$0.01968838 c = 0.42360231$

Step 8.4.2

Divide each term in $0.01968838 c = 0.42360231$ by $0.01968838$ and simplify.

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Step 8.4.2.1

Divide each term in $0.01968838 c = 0.42360231$ by $0.01968838$ .

$\frac{0.01968838 c}{0.01968838} = \frac{0.42360231}{0.01968838}$

Step 8.4.2.2

Simplify the left side.

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Step 8.4.2.2.1

Cancel the common factor of $0.01968838$ .

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Step 8.4.2.2.1.1

Cancel the common factor.

$\frac{0.01968838 c}{0.01968838} = \frac{0.42360231}{0.01968838}$

Step 8.4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{0.42360231}{0.01968838}$

Step 8.4.2.3

Simplify the right side.

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Step 8.4.2.3.1

Divide $0.42360231$ by $0.01968838$ .

$c = 21.51534362$

Step 9

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

$A = 16$

$B = 9.0622266$

$C = 154.93777339$

$a = 14$

$b = 8$

$c = 21.51534362$