Trigonometry Examples

$A = 71$ , $B = 62$ , $C = 47$ , $a = 20$

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 $b$ .

$\frac{\sin (62)}{b} = \frac{\sin (71)}{20}$

Step 3

Solve the equation for $b$ .

Tap for more steps...

Step 3.1

Factor each term.

Tap for more steps...

Step 3.1.1

Evaluate $\sin (62)$ .

$\frac{0.88294759}{b} = \frac{\sin (71)}{20}$

Step 3.1.2

Evaluate $\sin (71)$ .

$\frac{0.88294759}{b} = \frac{0.94551857}{20}$

Step 3.1.3

Divide $0.94551857$ by $20$ .

$\frac{0.88294759}{b} = 0.04727592$

Step 3.2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 3.2.1

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

$b, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$b$

Step 3.3

Multiply each term in $\frac{0.88294759}{b} = 0.04727592$ by $b$ to eliminate the fractions.

Tap for more steps...

Step 3.3.1

Multiply each term in $\frac{0.88294759}{b} = 0.04727592$ by $b$ .

$\frac{0.88294759}{b} b = 0.04727592 b$

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $b$ .

Tap for more steps...

Step 3.3.2.1.1

Cancel the common factor.

$\frac{0.88294759}{b} b = 0.04727592 b$

Step 3.3.2.1.2

Rewrite the expression.

$0.88294759 = 0.04727592 b$

Step 3.4

Solve the equation.

Tap for more steps...

Step 3.4.1

Rewrite the equation as $0.04727592 b = 0.88294759$ .

$0.04727592 b = 0.88294759$

Step 3.4.2

Divide each term in $0.04727592 b = 0.88294759$ by $0.04727592$ and simplify.

Tap for more steps...

Step 3.4.2.1

Divide each term in $0.04727592 b = 0.88294759$ by $0.04727592$ .

$\frac{0.04727592 b}{0.04727592} = \frac{0.88294759}{0.04727592}$

Step 3.4.2.2

Simplify the left side.

Tap for more steps...

Step 3.4.2.2.1

Cancel the common factor of $0.04727592$ .

Tap for more steps...

Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{0.04727592 b}{0.04727592} = \frac{0.88294759}{0.04727592}$

Step 3.4.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{0.88294759}{0.04727592}$

Step 3.4.2.3

Simplify the right side.

Tap for more steps...

Step 3.4.2.3.1

Divide $0.88294759$ by $0.04727592$ .

$b = 18.67647269$

Step 4

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

Step 5

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

$\frac{\sin (47)}{c} = \frac{\sin (71)}{20}$

Step 6

Solve the equation for $c$ .

Tap for more steps...

Step 6.1

Factor each term.

Tap for more steps...

Step 6.1.1

Evaluate $\sin (47)$ .

$\frac{0.7313537}{c} = \frac{\sin (71)}{20}$

Step 6.1.2

Evaluate $\sin (71)$ .

$\frac{0.7313537}{c} = \frac{0.94551857}{20}$

Step 6.1.3

Divide $0.94551857$ by $20$ .

$\frac{0.7313537}{c} = 0.04727592$

Step 6.2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 6.2.1

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

$c, 1$

Step 6.2.2

The LCM of one and any expression is the expression.

$c$

Step 6.3

Multiply each term in $\frac{0.7313537}{c} = 0.04727592$ by $c$ to eliminate the fractions.

Tap for more steps...

Step 6.3.1

Multiply each term in $\frac{0.7313537}{c} = 0.04727592$ by $c$ .

$\frac{0.7313537}{c} c = 0.04727592 c$

Step 6.3.2

Simplify the left side.

Tap for more steps...

Step 6.3.2.1

Cancel the common factor of $c$ .

Tap for more steps...

Step 6.3.2.1.1

Cancel the common factor.

$\frac{0.7313537}{c} c = 0.04727592 c$

Step 6.3.2.1.2

Rewrite the expression.

$0.7313537 = 0.04727592 c$

Step 6.4

Solve the equation.

Tap for more steps...

Step 6.4.1

Rewrite the equation as $0.04727592 c = 0.7313537$ .

$0.04727592 c = 0.7313537$

Step 6.4.2

Divide each term in $0.04727592 c = 0.7313537$ by $0.04727592$ and simplify.

Tap for more steps...

Step 6.4.2.1

Divide each term in $0.04727592 c = 0.7313537$ by $0.04727592$ .

$\frac{0.04727592 c}{0.04727592} = \frac{0.7313537}{0.04727592}$

Step 6.4.2.2

Simplify the left side.

Tap for more steps...

Step 6.4.2.2.1

Cancel the common factor of $0.04727592$ .

Tap for more steps...

Step 6.4.2.2.1.1

Cancel the common factor.

$\frac{0.04727592 c}{0.04727592} = \frac{0.7313537}{0.04727592}$

Step 6.4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{0.7313537}{0.04727592}$

Step 6.4.2.3

Simplify the right side.

Tap for more steps...

Step 6.4.2.3.1

Divide $0.7313537$ by $0.04727592$ .

$c = 15.469896$

Step 7

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

$A = 71$

$B = 62$

$C = 47$

$a = 20$

$b = 18.67647269$

$c = 15.469896$