Trigonometry Examples

$B = 85$ , $C = 15$ , $b = 40$

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 $c$ .

$\frac{\sin (15)}{c} = \frac{\sin (85)}{40}$

Step 3

Solve the equation for $c$ .

Tap for more steps...

Step 3.1

Factor each term.

Tap for more steps...

Step 3.1.1

The exact value of $\sin (15)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

Tap for more steps...

Step 3.1.1.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\frac{\sin (45 - 30)}{c} = \frac{\sin (85)}{40}$

Step 3.1.1.2

Separate negation.

$\frac{\sin (45 - (30))}{c} = \frac{\sin (85)}{40}$

Step 3.1.1.3

Apply the difference of angles identity.

$\frac{\sin (45) \cos (30) - \cos (45) \sin (30)}{c} = \frac{\sin (85)}{40}$

Step 3.1.1.4

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30)}{c} = \frac{\sin (85)}{40}$

Step 3.1.1.5

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30)}{c} = \frac{\sin (85)}{40}$

Step 3.1.1.6

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30)}{c} = \frac{\sin (85)}{40}$

Step 3.1.1.7

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$\frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{c} = \frac{\sin (85)}{40}$

Step 3.1.1.8

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 3.1.1.8.1

Simplify each term.

Tap for more steps...

Step 3.1.1.8.1.1

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

Tap for more steps...

Step 3.1.1.8.1.1.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{3}}{2}$ .

$\frac{\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{c} = \frac{\sin (85)}{40}$

Step 3.1.1.8.1.1.2

Combine using the product rule for radicals.

$\frac{\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{c} = \frac{\sin (85)}{40}$

Step 3.1.1.8.1.1.3

Multiply $2$ by $3$ .

$\frac{\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{c} = \frac{\sin (85)}{40}$

Step 3.1.1.8.1.1.4

Multiply $2$ by $2$ .

$\frac{\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{c} = \frac{\sin (85)}{40}$

Step 3.1.1.8.1.2

Multiply $- \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 3.1.1.8.1.2.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{2}}{2}$ .

$\frac{\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}}{c} = \frac{\sin (85)}{40}$

Step 3.1.1.8.1.2.2

Multiply $2$ by $2$ .

$\frac{\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}}{c} = \frac{\sin (85)}{40}$

Step 3.1.1.8.2

Combine the numerators over the common denominator.

$\frac{\frac{\sqrt{6} - \sqrt{2}}{4}}{c} = \frac{\sin (85)}{40}$

Step 3.1.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{1}{c} = \frac{\sin (85)}{40}$

Step 3.1.3

Multiply $\frac{\sqrt{6} - \sqrt{2}}{4}$ by $\frac{1}{c}$ .

$\frac{\sqrt{6} - \sqrt{2}}{4 c} = \frac{\sin (85)}{40}$

Step 3.1.4

Evaluate $\sin (85)$ .

$\frac{\sqrt{6} - \sqrt{2}}{4 c} = \frac{0.99619469}{40}$

Step 3.1.5

Divide $0.99619469$ by $40$ .

$\frac{\sqrt{6} - \sqrt{2}}{4 c} = 0.02490486$

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.

$4 c, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$4 c$

Step 3.3

Multiply each term in $\frac{\sqrt{6} - \sqrt{2}}{4 c} = 0.02490486$ by $4 c$ to eliminate the fractions.

Tap for more steps...

Step 3.3.1

Multiply each term in $\frac{\sqrt{6} - \sqrt{2}}{4 c} = 0.02490486$ by $4 c$ .

$\frac{\sqrt{6} - \sqrt{2}}{4 c} (4 c) = 0.02490486 (4 c)$

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Rewrite using the commutative property of multiplication.

$4 \frac{\sqrt{6} - \sqrt{2}}{4 c} c = 0.02490486 (4 c)$

Step 3.3.2.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 3.3.2.2.1

Factor $4$ out of $4 c$ .

$4 \frac{\sqrt{6} - \sqrt{2}}{4 (c)} c = 0.02490486 (4 c)$

Step 3.3.2.2.2

Cancel the common factor.

$4 \frac{\sqrt{6} - \sqrt{2}}{4 c} c = 0.02490486 (4 c)$

Step 3.3.2.2.3

Rewrite the expression.

$\frac{\sqrt{6} - \sqrt{2}}{c} c = 0.02490486 (4 c)$

Step 3.3.2.3

Cancel the common factor of $c$ .

Tap for more steps...

Step 3.3.2.3.1

Cancel the common factor.

$\frac{\sqrt{6} - \sqrt{2}}{c} c = 0.02490486 (4 c)$

Step 3.3.2.3.2

Rewrite the expression.

$\sqrt{6} - \sqrt{2} = 0.02490486 (4 c)$

Step 3.3.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.1

Multiply $4$ by $0.02490486$ .

$\sqrt{6} - \sqrt{2} = 0.09961946 c$

Step 3.4

Solve the equation.

Tap for more steps...

Step 3.4.1

Rewrite the equation as $0.09961946 c = \sqrt{6} - \sqrt{2}$ .

$0.09961946 c = \sqrt{6} - \sqrt{2}$

Step 3.4.2

Divide each term in $0.09961946 c = \sqrt{6} - \sqrt{2}$ by $0.09961946$ and simplify.

Tap for more steps...

Step 3.4.2.1

Divide each term in $0.09961946 c = \sqrt{6} - \sqrt{2}$ by $0.09961946$ .

$\frac{0.09961946 c}{0.09961946} = \frac{\sqrt{6}}{0.09961946} + \frac{- \sqrt{2}}{0.09961946}$

Step 3.4.2.2

Simplify the left side.

Tap for more steps...

Step 3.4.2.2.1

Cancel the common factor of $0.09961946$ .

Tap for more steps...

Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{0.09961946 c}{0.09961946} = \frac{\sqrt{6}}{0.09961946} + \frac{- \sqrt{2}}{0.09961946}$

Step 3.4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{\sqrt{6}}{0.09961946} + \frac{- \sqrt{2}}{0.09961946}$

Step 3.4.2.3

Simplify the right side.

Tap for more steps...

Step 3.4.2.3.1

Simplify each term.

Tap for more steps...

Step 3.4.2.3.1.1

Evaluate the root.

$c = \frac{2.44948974}{0.09961946} + \frac{- \sqrt{2}}{0.09961946}$

Step 3.4.2.3.1.2

Divide $2.44948974$ by $0.09961946$ .

$c = 24.58846395 + \frac{- \sqrt{2}}{0.09961946}$

Step 3.4.2.3.1.3

Move the negative in front of the fraction.

$c = 24.58846395 - \frac{\sqrt{2}}{0.09961946}$

Step 3.4.2.3.1.4

Evaluate the root.

$c = 24.58846395 - \frac{1.41421356}{0.09961946}$

Step 3.4.2.3.1.5

Divide $1.41421356$ by $0.09961946$ .

$c = 24.58846395 - 1 \cdot 14.19615628$

Step 3.4.2.3.1.6

Multiply $- 1$ by $14.19615628$ .

$c = 24.58846395 - 14.19615628$

Step 3.4.2.3.2

Subtract $14.19615628$ from $24.58846395$ .

$c = 10.39230767$

Step 4

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

$A + 15 + 85 = 180$

Step 5

Solve the equation for $A$ .

Tap for more steps...

Step 5.1

Add $15$ and $85$ .

$A + 100 = 180$

Step 5.2

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

Tap for more steps...

Step 5.2.1

Subtract $100$ from both sides of the equation.

$A = 180 - 100$

Step 5.2.2

Subtract $100$ from $180$ .

$A = 80$

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 $a$ .

$\frac{\sin (80)}{a} = \frac{\sin (85)}{40}$

Step 8

Solve the equation for $a$ .

Tap for more steps...

Step 8.1

Factor each term.

Tap for more steps...

Step 8.1.1

Evaluate $\sin (80)$ .

$\frac{0.98480775}{a} = \frac{\sin (85)}{40}$

Step 8.1.2

Evaluate $\sin (85)$ .

$\frac{0.98480775}{a} = \frac{0.99619469}{40}$

Step 8.1.3

Divide $0.99619469$ by $40$ .

$\frac{0.98480775}{a} = 0.02490486$

Step 8.2

Find the LCD of the terms in the equation.

Tap for more steps...

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.

$a, 1$

Step 8.2.2

The LCM of one and any expression is the expression.

$a$

Step 8.3

Multiply each term in $\frac{0.98480775}{a} = 0.02490486$ by $a$ to eliminate the fractions.

Tap for more steps...

Step 8.3.1

Multiply each term in $\frac{0.98480775}{a} = 0.02490486$ by $a$ .

$\frac{0.98480775}{a} a = 0.02490486 a$

Step 8.3.2

Simplify the left side.

Tap for more steps...

Step 8.3.2.1

Cancel the common factor of $a$ .

Tap for more steps...

Step 8.3.2.1.1

Cancel the common factor.

$\frac{0.98480775}{a} a = 0.02490486 a$

Step 8.3.2.1.2

Rewrite the expression.

$0.98480775 = 0.02490486 a$

Step 8.4

Solve the equation.

Tap for more steps...

Step 8.4.1

Rewrite the equation as $0.02490486 a = 0.98480775$ .

$0.02490486 a = 0.98480775$

Step 8.4.2

Divide each term in $0.02490486 a = 0.98480775$ by $0.02490486$ and simplify.

Tap for more steps...

Step 8.4.2.1

Divide each term in $0.02490486 a = 0.98480775$ by $0.02490486$ .

$\frac{0.02490486 a}{0.02490486} = \frac{0.98480775}{0.02490486}$

Step 8.4.2.2

Simplify the left side.

Tap for more steps...

Step 8.4.2.2.1

Cancel the common factor of $0.02490486$ .

Tap for more steps...

Step 8.4.2.2.1.1

Cancel the common factor.

$\frac{0.02490486 a}{0.02490486} = \frac{0.98480775}{0.02490486}$

Step 8.4.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{0.98480775}{0.02490486}$

Step 8.4.2.3

Simplify the right side.

Tap for more steps...

Step 8.4.2.3.1

Divide $0.98480775$ by $0.02490486$ .

$a = 39.54278234$

Step 9

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

$A = 80$

$B = 85$

$C = 15$

$a = 39.54278234$

$b = 40$

$c = 10.39230767$