Trigonometry Examples

$C = 41$ , $B = 105$ , $b = 12.6$

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 $c$ .

$\frac{\sin (41)}{c} = \frac{\sin (105)}{12.6}$

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

Evaluate $\sin (41)$ .

$\frac{0.65605902}{c} = \frac{\sin (105)}{12.6}$

Step 3.1.2

The exact value of $\sin (105)$ is $\frac{\sqrt{2} + \sqrt{6}}{4}$ .

Tap for more steps...

Step 3.1.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{0.65605902}{c} = \frac{\sin (75)}{12.6}$

Step 3.1.2.2

Split $75$ into two angles where the values of the six trigonometric functions are known.

$\frac{0.65605902}{c} = \frac{\sin (30 + 45)}{12.6}$

Step 3.1.2.3

Apply the sum of angles identity.

$\frac{0.65605902}{c} = \frac{\sin (30) \cos (45) + \cos (30) \sin (45)}{12.6}$

Step 3.1.2.4

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

$\frac{0.65605902}{c} = \frac{\frac{1}{2} \cos (45) + \cos (30) \sin (45)}{12.6}$

Step 3.1.2.5

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

$\frac{0.65605902}{c} = \frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45)}{12.6}$

Step 3.1.2.6

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

$\frac{0.65605902}{c} = \frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \sin (45)}{12.6}$

Step 3.1.2.7

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

$\frac{0.65605902}{c} = \frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{12.6}$

Step 3.1.2.8

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

Tap for more steps...

Step 3.1.2.8.1

Simplify each term.

Tap for more steps...

Step 3.1.2.8.1.1

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

Tap for more steps...

Step 3.1.2.8.1.1.1

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

$\frac{0.65605902}{c} = \frac{\frac{\sqrt{2}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{12.6}$

Step 3.1.2.8.1.1.2

Multiply $2$ by $2$ .

$\frac{0.65605902}{c} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{12.6}$

Step 3.1.2.8.1.2

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

Tap for more steps...

Step 3.1.2.8.1.2.1

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

$\frac{0.65605902}{c} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}}{12.6}$

Step 3.1.2.8.1.2.2

Combine using the product rule for radicals.

$\frac{0.65605902}{c} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}}{12.6}$

Step 3.1.2.8.1.2.3

Multiply $3$ by $2$ .

$\frac{0.65605902}{c} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{2 \cdot 2}}{12.6}$

Step 3.1.2.8.1.2.4

Multiply $2$ by $2$ .

$\frac{0.65605902}{c} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}{12.6}$

Step 3.1.2.8.2

Combine the numerators over the common denominator.

$\frac{0.65605902}{c} = \frac{\frac{\sqrt{2} + \sqrt{6}}{4}}{12.6}$

Step 3.1.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{0.65605902}{c} = \frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{1}{12.6}$

Step 3.1.4

Divide $1$ by $12.6$ .

$\frac{0.65605902}{c} = \frac{\sqrt{2} + \sqrt{6}}{4} \cdot 0.\overline{079365}$

Step 3.1.5

Multiply $\frac{\sqrt{2} + \sqrt{6}}{4} \cdot 0.\overline{079365}$ .

Tap for more steps...

Step 3.1.5.1

Combine $\frac{\sqrt{2} + \sqrt{6}}{4}$ and $0.\overline{079365}$ .

$\frac{0.65605902}{c} = \frac{(\sqrt{2} + \sqrt{6}) \cdot 0.\overline{079365}}{4}$

Step 3.1.5.2

Multiply $\sqrt{2} + \sqrt{6}$ by $0.\overline{079365}$ .

$\frac{0.65605902}{c} = \frac{0.30664311}{4}$

Step 3.1.6

Divide $0.30664311$ by $4$ .

$\frac{0.65605902}{c} = 0.07666077$

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.

$c, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$c$

Step 3.3

Multiply each term in $\frac{0.65605902}{c} = 0.07666077$ by $c$ to eliminate the fractions.

Tap for more steps...

Step 3.3.1

Multiply each term in $\frac{0.65605902}{c} = 0.07666077$ by $c$ .

$\frac{0.65605902}{c} c = 0.07666077 c$

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $c$ .

Tap for more steps...

Step 3.3.2.1.1

Cancel the common factor.

$\frac{0.65605902}{c} c = 0.07666077 c$

Step 3.3.2.1.2

Rewrite the expression.

$0.65605902 = 0.07666077 c$

Step 3.4

Solve the equation.

Tap for more steps...

Step 3.4.1

Rewrite the equation as $0.07666077 c = 0.65605902$ .

$0.07666077 c = 0.65605902$

Step 3.4.2

Divide each term in $0.07666077 c = 0.65605902$ by $0.07666077$ and simplify.

Tap for more steps...

Step 3.4.2.1

Divide each term in $0.07666077 c = 0.65605902$ by $0.07666077$ .

$\frac{0.07666077 c}{0.07666077} = \frac{0.65605902}{0.07666077}$

Step 3.4.2.2

Simplify the left side.

Tap for more steps...

Step 3.4.2.2.1

Cancel the common factor of $0.07666077$ .

Tap for more steps...

Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{0.07666077 c}{0.07666077} = \frac{0.65605902}{0.07666077}$

Step 3.4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{0.65605902}{0.07666077}$

Step 3.4.2.3

Simplify the right side.

Tap for more steps...

Step 3.4.2.3.1

Divide $0.65605902$ by $0.07666077$ .

$c = 8.55794879$

Step 4

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

$A + 41 + 105 = 180$

Step 5

Solve the equation for $A$ .

Tap for more steps...

Step 5.1

Add $41$ and $105$ .

$A + 146 = 180$

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 $146$ from both sides of the equation.

$A = 180 - 146$

Step 5.2.2

Subtract $146$ from $180$ .

$A = 34$

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 $a$ .

$\frac{\sin (34)}{a} = \frac{\sin (105)}{12.6}$

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 $\sin (34)$ .

$\frac{0.5591929}{a} = \frac{\sin (105)}{12.6}$

Step 8.1.2

The exact value of $\sin (105)$ is $\frac{\sqrt{2} + \sqrt{6}}{4}$ .

Tap for more steps...

Step 8.1.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{0.5591929}{a} = \frac{\sin (75)}{12.6}$

Step 8.1.2.2

Split $75$ into two angles where the values of the six trigonometric functions are known.

$\frac{0.5591929}{a} = \frac{\sin (30 + 45)}{12.6}$

Step 8.1.2.3

Apply the sum of angles identity.

$\frac{0.5591929}{a} = \frac{\sin (30) \cos (45) + \cos (30) \sin (45)}{12.6}$

Step 8.1.2.4

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

$\frac{0.5591929}{a} = \frac{\frac{1}{2} \cos (45) + \cos (30) \sin (45)}{12.6}$

Step 8.1.2.5

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

$\frac{0.5591929}{a} = \frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45)}{12.6}$

Step 8.1.2.6

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

$\frac{0.5591929}{a} = \frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \sin (45)}{12.6}$

Step 8.1.2.7

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

$\frac{0.5591929}{a} = \frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{12.6}$

Step 8.1.2.8

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

Tap for more steps...

Step 8.1.2.8.1

Simplify each term.

Tap for more steps...

Step 8.1.2.8.1.1

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

Tap for more steps...

Step 8.1.2.8.1.1.1

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

$\frac{0.5591929}{a} = \frac{\frac{\sqrt{2}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{12.6}$

Step 8.1.2.8.1.1.2

Multiply $2$ by $2$ .

$\frac{0.5591929}{a} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{12.6}$

Step 8.1.2.8.1.2

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

Tap for more steps...

Step 8.1.2.8.1.2.1

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

$\frac{0.5591929}{a} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}}{12.6}$

Step 8.1.2.8.1.2.2

Combine using the product rule for radicals.

$\frac{0.5591929}{a} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}}{12.6}$

Step 8.1.2.8.1.2.3

Multiply $3$ by $2$ .

$\frac{0.5591929}{a} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{2 \cdot 2}}{12.6}$

Step 8.1.2.8.1.2.4

Multiply $2$ by $2$ .

$\frac{0.5591929}{a} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}{12.6}$

Step 8.1.2.8.2

Combine the numerators over the common denominator.

$\frac{0.5591929}{a} = \frac{\frac{\sqrt{2} + \sqrt{6}}{4}}{12.6}$

Step 8.1.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{0.5591929}{a} = \frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{1}{12.6}$

Step 8.1.4

Divide $1$ by $12.6$ .

$\frac{0.5591929}{a} = \frac{\sqrt{2} + \sqrt{6}}{4} \cdot 0.\overline{079365}$

Step 8.1.5

Multiply $\frac{\sqrt{2} + \sqrt{6}}{4} \cdot 0.\overline{079365}$ .

Tap for more steps...

Step 8.1.5.1

Combine $\frac{\sqrt{2} + \sqrt{6}}{4}$ and $0.\overline{079365}$ .

$\frac{0.5591929}{a} = \frac{(\sqrt{2} + \sqrt{6}) \cdot 0.\overline{079365}}{4}$

Step 8.1.5.2

Multiply $\sqrt{2} + \sqrt{6}$ by $0.\overline{079365}$ .

$\frac{0.5591929}{a} = \frac{0.30664311}{4}$

Step 8.1.6

Divide $0.30664311$ by $4$ .

$\frac{0.5591929}{a} = 0.07666077$

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 $\frac{0.5591929}{a} = 0.07666077$ by $a$ to eliminate the fractions.

Tap for more steps...

Step 8.3.1

Multiply each term in $\frac{0.5591929}{a} = 0.07666077$ by $a$ .

$\frac{0.5591929}{a} a = 0.07666077 a$

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.

$\frac{0.5591929}{a} a = 0.07666077 a$

Step 8.3.2.1.2

Rewrite the expression.

$0.5591929 = 0.07666077 a$

Step 8.4

Solve the equation.

Tap for more steps...

Step 8.4.1

Rewrite the equation as $0.07666077 a = 0.5591929$ .

$0.07666077 a = 0.5591929$

Step 8.4.2

Divide each term in $0.07666077 a = 0.5591929$ by $0.07666077$ and simplify.

Tap for more steps...

Step 8.4.2.1

Divide each term in $0.07666077 a = 0.5591929$ by $0.07666077$ .

$\frac{0.07666077 a}{0.07666077} = \frac{0.5591929}{0.07666077}$

Step 8.4.2.2

Simplify the left side.

Tap for more steps...

Step 8.4.2.2.1

Cancel the common factor of $0.07666077$ .

Tap for more steps...

Step 8.4.2.2.1.1

Cancel the common factor.

$\frac{0.07666077 a}{0.07666077} = \frac{0.5591929}{0.07666077}$

Step 8.4.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{0.5591929}{0.07666077}$

Step 8.4.2.3

Simplify the right side.

Tap for more steps...

Step 8.4.2.3.1

Divide $0.5591929$ by $0.07666077$ .

$a = 7.29438057$

Step 9

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

$A = 34$

$B = 105$

$C = 41$

$a = 7.29438057$

$b = 12.6$

$c = 8.55794879$