Trigonometry Examples

$B = 40$ , $b = 25$ , $c = 26$

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 $C$ .

$\frac{\sin (C)}{26} = \frac{\sin (40)}{25}$

Step 4

Solve the equation for $C$ .

Tap for more steps...

Step 4.1

Multiply both sides of the equation by $26$ .

$26 \frac{\sin (C)}{26} = 26 \frac{\sin (40)}{25}$

Step 4.2

Simplify both sides of the equation.

Tap for more steps...

Step 4.2.1

Simplify the left side.

Tap for more steps...

Step 4.2.1.1

Cancel the common factor of $26$ .

Tap for more steps...

Step 4.2.1.1.1

Cancel the common factor.

$26 \frac{\sin (C)}{26} = 26 \frac{\sin (40)}{25}$

Step 4.2.1.1.2

Rewrite the expression.

$\sin (C) = 26 \frac{\sin (40)}{25}$

Step 4.2.2

Simplify the right side.

Tap for more steps...

Step 4.2.2.1

Simplify $26 \frac{\sin (40)}{25}$ .

Tap for more steps...

Step 4.2.2.1.1

Evaluate $\sin (40)$ .

$\sin (C) = 26 (\frac{0.6427876}{25})$

Step 4.2.2.1.2

Divide $0.6427876$ by $25$ .

$\sin (C) = 26 \cdot 0.0257115$

Step 4.2.2.1.3

Multiply $26$ by $0.0257115$ .

$\sin (C) = 0.66849911$

Step 4.3

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

$C = \arcsin (0.66849911)$

Step 4.4

Simplify the right side.

Tap for more steps...

Step 4.4.1

Evaluate $\arcsin (0.66849911)$ .

$C = 41.95133113$

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.

$C = 180 - 41.95133113$

Step 4.6

Subtract $41.95133113$ from $180$ .

$C = 138.04866886$

Step 4.7

The solution to the equation $C = 41.95133113$ .

$C = 41.95133113, 138.04866886$

Step 5

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

$A + 41.95133113 + 40 = 180$

Step 6

Solve the equation for $A$ .

Tap for more steps...

Step 6.1

Add $41.95133113$ and $40$ .

$A + 81.95133113 = 180$

Step 6.2

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

Tap for more steps...

Step 6.2.1

Subtract $81.95133113$ from both sides of the equation.

$A = 180 - 81.95133113$

Step 6.2.2

Subtract $81.95133113$ from $180$ .

$A = 98.04866886$

Step 7

Use the law of cosines to find the unknown side of the triangle, given the other two sides and the included angle.

$a^{2} = b^{2} + c^{2} - 2 b c \cos (A)$

Step 8

Solve the equation.

$a = \sqrt{b^{2} + c^{2} - 2 b c \cos (A)}$

Step 9

Substitute the known values into the equation.

$a = \sqrt{{(25)}^{2} + {(26)}^{2} - 2 \cdot 25 \cdot 26 \cos (98.04866886)}$

Step 10

Simplify the results.

Tap for more steps...

Step 10.1

Raise $25$ to the power of $2$ .

$a = \sqrt{625 + {(26)}^{2} - 2 \cdot 25 \cdot (26 \cos (98.04866886))}$

Step 10.2

Raise $26$ to the power of $2$ .

$a = \sqrt{625 + 676 - 2 \cdot 25 \cdot (26 \cos (98.04866886))}$

Step 10.3

Multiply $- 2 \cdot 25 \cdot 26$ .

Tap for more steps...

Step 10.3.1

Multiply $- 2$ by $25$ .

$a = \sqrt{625 + 676 - 50 \cdot (26 \cos (98.04866886))}$

Step 10.3.2

Multiply $- 50$ by $26$ .

$a = \sqrt{625 + 676 - 1300 \cos (98.04866886)}$

Step 10.4

Add $625$ and $676$ .

$a = \sqrt{1301 - 1300 \cos (98.04866886)}$

Step 11

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

Solve for the second triangle.

Step 12

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

Step 13

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

$\frac{\sin (C)}{26} = \frac{\sin (40)}{25}$

Step 14

Solve the equation for $C$ .

Tap for more steps...

Step 14.1

Multiply both sides of the equation by $26$ .

$26 \frac{\sin (C)}{26} = 26 \frac{\sin (40)}{25}$

Step 14.2

Simplify both sides of the equation.

Tap for more steps...

Step 14.2.1

Simplify the left side.

Tap for more steps...

Step 14.2.1.1

Cancel the common factor of $26$ .

Tap for more steps...

Step 14.2.1.1.1

Cancel the common factor.

$26 \frac{\sin (C)}{26} = 26 \frac{\sin (40)}{25}$

Step 14.2.1.1.2

Rewrite the expression.

$\sin (C) = 26 \frac{\sin (40)}{25}$

Step 14.2.2

Simplify the right side.

Tap for more steps...

Step 14.2.2.1

Simplify $26 \frac{\sin (40)}{25}$ .

Tap for more steps...

Step 14.2.2.1.1

Evaluate $\sin (40)$ .

$\sin (C) = 26 (\frac{0.6427876}{25})$

Step 14.2.2.1.2

Divide $0.6427876$ by $25$ .

$\sin (C) = 26 \cdot 0.0257115$

Step 14.2.2.1.3

Multiply $26$ by $0.0257115$ .

$\sin (C) = 0.66849911$

Step 14.3

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

$C = \arcsin (0.66849911)$

Step 14.4

Simplify the right side.

Tap for more steps...

Step 14.4.1

Evaluate $\arcsin (0.66849911)$ .

$C = 41.95133113$

Step 14.5

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.

$C = 180 - 41.95133113$

Step 14.6

Subtract $41.95133113$ from $180$ .

$C = 138.04866886$

Step 14.7

The solution to the equation $C = 41.95133113$ .

$C = 41.95133113, 138.04866886$

Step 15

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

$A + 138.04866886 + 40 = 180$

Step 16

Solve the equation for $A$ .

Tap for more steps...

Step 16.1

Add $138.04866886$ and $40$ .

$A + 178.04866886 = 180$

Step 16.2

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

Tap for more steps...

Step 16.2.1

Subtract $178.04866886$ from both sides of the equation.

$A = 180 - 178.04866886$

Step 16.2.2

Subtract $178.04866886$ from $180$ .

$A = 1.95133113$

Step 17

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

Step 18

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

$\frac{\sin (1.95133113)}{a} = \frac{\sin (40)}{25}$

Step 19

Solve the equation for $a$ .

Tap for more steps...

Step 19.1

Factor each term.

Tap for more steps...

Step 19.1.1

Evaluate $\sin (1.95133113)$ .

$\frac{0.03405056}{a} = \frac{\sin (40)}{25}$

Step 19.1.2

Evaluate $\sin (40)$ .

$\frac{0.03405056}{a} = \frac{0.6427876}{25}$

Step 19.1.3

Divide $0.6427876$ by $25$ .

$\frac{0.03405056}{a} = 0.0257115$

Step 19.2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 19.2.1

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

$a, 1$

Step 19.2.2

The LCM of one and any expression is the expression.

$a$

Step 19.3

Multiply each term in $\frac{0.03405056}{a} = 0.0257115$ by $a$ to eliminate the fractions.

Tap for more steps...

Step 19.3.1

Multiply each term in $\frac{0.03405056}{a} = 0.0257115$ by $a$ .

$\frac{0.03405056}{a} a = 0.0257115 a$

Step 19.3.2

Simplify the left side.

Tap for more steps...

Step 19.3.2.1

Cancel the common factor of $a$ .

Tap for more steps...

Step 19.3.2.1.1

Cancel the common factor.

$\frac{0.03405056}{a} a = 0.0257115 a$

Step 19.3.2.1.2

Rewrite the expression.

$0.03405056 = 0.0257115 a$

Step 19.4

Solve the equation.

Tap for more steps...

Step 19.4.1

Rewrite the equation as $0.0257115 a = 0.03405056$ .

$0.0257115 a = 0.03405056$

Step 19.4.2

Divide each term in $0.0257115 a = 0.03405056$ by $0.0257115$ and simplify.

Tap for more steps...

Step 19.4.2.1

Divide each term in $0.0257115 a = 0.03405056$ by $0.0257115$ .

$\frac{0.0257115 a}{0.0257115} = \frac{0.03405056}{0.0257115}$

Step 19.4.2.2

Simplify the left side.

Tap for more steps...

Step 19.4.2.2.1

Cancel the common factor of $0.0257115$ .

Tap for more steps...

Step 19.4.2.2.1.1

Cancel the common factor.

$\frac{0.0257115 a}{0.0257115} = \frac{0.03405056}{0.0257115}$

Step 19.4.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{0.03405056}{0.0257115}$

Step 19.4.2.3

Simplify the right side.

Tap for more steps...

Step 19.4.2.3.1

Divide $0.03405056$ by $0.0257115$ .

$a = 1.32433206$

Step 20

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

First Triangle Combination:

$A = 98.04866886$

$B = 40$

$C = 41.95133113$

$a = \sqrt{1301 - 1300 \cos (98.04866886)}$

$b = 25$

$c = 26$

Second Triangle Combination:

$A = 1.95133113$

$B = 40$

$C = 138.04866886$

$a = 1.32433206$

$b = 25$

$c = 26$