Trigonometry Examples

$A = 15$ , $a = 4$ , $b = 5$

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 $B$ .

$\frac{\sin (B)}{5} = \frac{\sin (15)}{4}$

Step 4

Solve the equation for $B$ .

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Step 4.1

Multiply both sides of the equation by $5$ .

$5 \frac{\sin (B)}{5} = 5 \frac{\sin (15)}{4}$

Step 4.2

Simplify both sides of the equation.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Cancel the common factor of $5$ .

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Step 4.2.1.1.1

Cancel the common factor.

$5 \frac{\sin (B)}{5} = 5 \frac{\sin (15)}{4}$

Step 4.2.1.1.2

Rewrite the expression.

$\sin (B) = 5 \frac{\sin (15)}{4}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $5 \frac{\sin (15)}{4}$ .

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Step 4.2.2.1.1

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

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Step 4.2.2.1.1.1

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

$\sin (B) = 5 \frac{\sin (45 - 30)}{4}$

Step 4.2.2.1.1.2

Separate negation.

$\sin (B) = 5 \frac{\sin (45 - (30))}{4}$

Step 4.2.2.1.1.3

Apply the difference of angles identity.

$\sin (B) = 5 \frac{\sin (45) \cos (30) - \cos (45) \sin (30)}{4}$

Step 4.2.2.1.1.4

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

$\sin (B) = 5 \frac{\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30)}{4}$

Step 4.2.2.1.1.5

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

$\sin (B) = 5 \frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30)}{4}$

Step 4.2.2.1.1.6

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

$\sin (B) = 5 \frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30)}{4}$

Step 4.2.2.1.1.7

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

$\sin (B) = 5 \frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{4}$

Step 4.2.2.1.1.8

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

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Step 4.2.2.1.1.8.1

Simplify each term.

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Step 4.2.2.1.1.8.1.1

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

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Step 4.2.2.1.1.8.1.1.1

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

$\sin (B) = 5 \frac{\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{4}$

Step 4.2.2.1.1.8.1.1.2

Combine using the product rule for radicals.

$\sin (B) = 5 \frac{\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{4}$

Step 4.2.2.1.1.8.1.1.3

Multiply $2$ by $3$ .

$\sin (B) = 5 \frac{\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{4}$

Step 4.2.2.1.1.8.1.1.4

Multiply $2$ by $2$ .

$\sin (B) = 5 \frac{\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{4}$

Step 4.2.2.1.1.8.1.2

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

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Step 4.2.2.1.1.8.1.2.1

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

$\sin (B) = 5 \frac{\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}}{4}$

Step 4.2.2.1.1.8.1.2.2

Multiply $2$ by $2$ .

$\sin (B) = 5 \frac{\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}}{4}$

Step 4.2.2.1.1.8.2

Combine the numerators over the common denominator.

$\sin (B) = 5 \frac{\frac{\sqrt{6} - \sqrt{2}}{4}}{4}$

Step 4.2.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$\sin (B) = 5 (\frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{1}{4})$

Step 4.2.2.1.3

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

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Step 4.2.2.1.3.1

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

$\sin (B) = 5 \frac{\sqrt{6} - \sqrt{2}}{4 \cdot 4}$

Step 4.2.2.1.3.2

Multiply $4$ by $4$ .

$\sin (B) = 5 \frac{\sqrt{6} - \sqrt{2}}{16}$

Step 4.2.2.1.4

Combine $5$ and $\frac{\sqrt{6} - \sqrt{2}}{16}$ .

$\sin (B) = \frac{5 (\sqrt{6} - \sqrt{2})}{16}$

Step 4.3

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

$B = \arcsin (\frac{5 (\sqrt{6} - \sqrt{2})}{16})$

Step 4.4

Simplify the right side.

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Step 4.4.1

Evaluate $\arcsin (\frac{5 (\sqrt{6} - \sqrt{2})}{16})$ .

$B = 18.87616421$

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.

$B = 180 - 18.87616421$

Step 4.6

Subtract $18.87616421$ from $180$ .

$B = 161.12383578$

Step 4.7

The solution to the equation $B = 18.87616421$ .

$B = 18.87616421, 161.12383578$

Step 5

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

$15 + C + 18.87616421 = 180$

Step 6

Solve the equation for $C$ .

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Step 6.1

Add $15$ and $18.87616421$ .

$C + 33.87616421 = 180$

Step 6.2

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

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Step 6.2.1

Subtract $33.87616421$ from both sides of the equation.

$C = 180 - 33.87616421$

Step 6.2.2

Subtract $33.87616421$ from $180$ .

$C = 146.12383578$

Step 7

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

Step 8

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

$\frac{\sin (146.12383578)}{c} = \frac{\sin (15)}{4}$

Step 9

Solve the equation for $c$ .

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Step 9.1

Factor each term.

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Step 9.1.1

Evaluate $\sin (146.12383578)$ .

$\frac{0.55739976}{c} = \frac{\sin (15)}{4}$

Step 9.1.2

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

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Step 9.1.2.1

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

$\frac{0.55739976}{c} = \frac{\sin (45 - 30)}{4}$

Step 9.1.2.2

Separate negation.

$\frac{0.55739976}{c} = \frac{\sin (45 - (30))}{4}$

Step 9.1.2.3

Apply the difference of angles identity.

$\frac{0.55739976}{c} = \frac{\sin (45) \cos (30) - \cos (45) \sin (30)}{4}$

Step 9.1.2.4

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

$\frac{0.55739976}{c} = \frac{\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30)}{4}$

Step 9.1.2.5

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

$\frac{0.55739976}{c} = \frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30)}{4}$

Step 9.1.2.6

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

$\frac{0.55739976}{c} = \frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30)}{4}$

Step 9.1.2.7

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

$\frac{0.55739976}{c} = \frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{4}$

Step 9.1.2.8

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

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Step 9.1.2.8.1

Simplify each term.

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Step 9.1.2.8.1.1

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

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Step 9.1.2.8.1.1.1

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

$\frac{0.55739976}{c} = \frac{\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{4}$

Step 9.1.2.8.1.1.2

Combine using the product rule for radicals.

$\frac{0.55739976}{c} = \frac{\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{4}$

Step 9.1.2.8.1.1.3

Multiply $2$ by $3$ .

$\frac{0.55739976}{c} = \frac{\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{4}$

Step 9.1.2.8.1.1.4

Multiply $2$ by $2$ .

$\frac{0.55739976}{c} = \frac{\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{4}$

Step 9.1.2.8.1.2

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

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Step 9.1.2.8.1.2.1

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

$\frac{0.55739976}{c} = \frac{\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}}{4}$

Step 9.1.2.8.1.2.2

Multiply $2$ by $2$ .

$\frac{0.55739976}{c} = \frac{\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}}{4}$

Step 9.1.2.8.2

Combine the numerators over the common denominator.

$\frac{0.55739976}{c} = \frac{\frac{\sqrt{6} - \sqrt{2}}{4}}{4}$

Step 9.1.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{0.55739976}{c} = \frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{1}{4}$

Step 9.1.4

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

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Step 9.1.4.1

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

$\frac{0.55739976}{c} = \frac{\sqrt{6} - \sqrt{2}}{4 \cdot 4}$

Step 9.1.4.2

Multiply $4$ by $4$ .

$\frac{0.55739976}{c} = \frac{\sqrt{6} - \sqrt{2}}{16}$

Step 9.2

Find the LCD of the terms in the equation.

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Step 9.2.1

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

$c, 16$

Step 9.2.2

Since $c, 16$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 16$ then find LCM for the variable part $c^{1}$ .

Step 9.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 9.2.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 9.2.5

The prime factors for $16$ are $2 \cdot 2 \cdot 2 \cdot 2$ .

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Step 9.2.5.1

$16$ has factors of $2$ and $8$ .

$2 \cdot 8$

Step 9.2.5.2

$8$ has factors of $2$ and $4$ .

$2 \cdot 2 \cdot 4$

Step 9.2.5.3

$4$ has factors of $2$ and $2$ .

$2 \cdot 2 \cdot 2 \cdot 2$

Step 9.2.6

Multiply $2 \cdot 2 \cdot 2 \cdot 2$ .

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Step 9.2.6.1

Multiply $2$ by $2$ .

$4 \cdot 2 \cdot 2$

Step 9.2.6.2

Multiply $4$ by $2$ .

$8 \cdot 2$

Step 9.2.6.3

Multiply $8$ by $2$ .

$16$

Step 9.2.7

The factor for $c^{1}$ is $c$ itself.

$c^{1} = c$

$c$ occurs $1$ time.

Step 9.2.8

The LCM of $c^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$c$

Step 9.2.9

The LCM for $c, 16$ is the numeric part $16$ multiplied by the variable part.

$16 c$

Step 9.3

Multiply each term in $\frac{0.55739976}{c} = \frac{\sqrt{6} - \sqrt{2}}{16}$ by $16 c$ to eliminate the fractions.

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Step 9.3.1

Multiply each term in $\frac{0.55739976}{c} = \frac{\sqrt{6} - \sqrt{2}}{16}$ by $16 c$ .

$\frac{0.55739976}{c} (16 c) = \frac{\sqrt{6} - \sqrt{2}}{16} (16 c)$

Step 9.3.2

Simplify the left side.

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Step 9.3.2.1

Rewrite using the commutative property of multiplication.

$16 \frac{0.55739976}{c} c = \frac{\sqrt{6} - \sqrt{2}}{16} (16 c)$

Step 9.3.2.2

Multiply $16 \frac{0.55739976}{c}$ .

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Step 9.3.2.2.1

Combine $16$ and $\frac{0.55739976}{c}$ .

$\frac{16 \cdot 0.55739976}{c} c = \frac{\sqrt{6} - \sqrt{2}}{16} (16 c)$

Step 9.3.2.2.2

Multiply $16$ by $0.55739976$ .

$\frac{8.91839623}{c} c = \frac{\sqrt{6} - \sqrt{2}}{16} (16 c)$

Step 9.3.2.3

Cancel the common factor of $c$ .

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Step 9.3.2.3.1

Cancel the common factor.

$\frac{8.91839623}{c} c = \frac{\sqrt{6} - \sqrt{2}}{16} (16 c)$

Step 9.3.2.3.2

Rewrite the expression.

$8.91839623 = \frac{\sqrt{6} - \sqrt{2}}{16} (16 c)$

Step 9.3.3

Simplify the right side.

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Step 9.3.3.1

Cancel the common factor of $16$ .

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Step 9.3.3.1.1

Factor $16$ out of $16 c$ .

$8.91839623 = \frac{\sqrt{6} - \sqrt{2}}{16} (16 (c))$

Step 9.3.3.1.2

Cancel the common factor.

$8.91839623 = \frac{\sqrt{6} - \sqrt{2}}{16} (16 c)$

Step 9.3.3.1.3

Rewrite the expression.

$8.91839623 = (\sqrt{6} - \sqrt{2}) c$

Step 9.3.3.2

Apply the distributive property.

$8.91839623 = \sqrt{6} c - \sqrt{2} c$

Step 9.4

Solve the equation.

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Step 9.4.1

Rewrite the equation as $\sqrt{6} c - \sqrt{2} c = 8.91839623$ .

$\sqrt{6} c - \sqrt{2} c = 8.91839623$

Step 9.4.2

Factor $c$ out of $\sqrt{6} c - \sqrt{2} c$ .

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Step 9.4.2.1

Factor $c$ out of $\sqrt{6} c$ .

$c \sqrt{6} - \sqrt{2} c = 8.91839623$

Step 9.4.2.2

Factor $c$ out of $- \sqrt{2} c$ .

$c \sqrt{6} + c (- \sqrt{2}) = 8.91839623$

Step 9.4.2.3

Factor $c$ out of $c \sqrt{6} + c (- \sqrt{2})$ .

$c (\sqrt{6} - \sqrt{2}) = 8.91839623$

Step 9.4.3

Divide each term in $c (\sqrt{6} - \sqrt{2}) = 8.91839623$ by $\sqrt{6} - \sqrt{2}$ and simplify.

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Step 9.4.3.1

Divide each term in $c (\sqrt{6} - \sqrt{2}) = 8.91839623$ by $\sqrt{6} - \sqrt{2}$ .

$\frac{c (\sqrt{6} - \sqrt{2})}{\sqrt{6} - \sqrt{2}} = \frac{8.91839623}{\sqrt{6} - \sqrt{2}}$

Step 9.4.3.2

Simplify the left side.

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Step 9.4.3.2.1

Cancel the common factor of $\sqrt{6} - \sqrt{2}$ .

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Step 9.4.3.2.1.1

Cancel the common factor.

$\frac{c (\sqrt{6} - \sqrt{2})}{\sqrt{6} - \sqrt{2}} = \frac{8.91839623}{\sqrt{6} - \sqrt{2}}$

Step 9.4.3.2.1.2

Divide $c$ by $1$ .

$c = \frac{8.91839623}{\sqrt{6} - \sqrt{2}}$

Step 9.4.3.3

Simplify the right side.

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Step 9.4.3.3.1

Multiply $\frac{8.91839623}{\sqrt{6} - \sqrt{2}}$ by $\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}$ .

$c = \frac{8.91839623}{\sqrt{6} - \sqrt{2}} \cdot \frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}$

Step 9.4.3.3.2

Multiply $\frac{8.91839623}{\sqrt{6} - \sqrt{2}}$ by $\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}$ .

$c = \frac{8.91839623 (\sqrt{6} + \sqrt{2})}{(\sqrt{6} - \sqrt{2}) (\sqrt{6} + \sqrt{2})}$

Step 9.4.3.3.3

Expand the denominator using the FOIL method.

$c = \frac{8.91839623 (\sqrt{6} + \sqrt{2})}{{\sqrt{6}}^{2} + \sqrt{12} - \sqrt{12} - {\sqrt{2}}^{2}}$

Step 9.4.3.3.4

Simplify.

$c = \frac{8.91839623 (\sqrt{6} + \sqrt{2})}{4}$

Step 9.4.3.3.5

Multiply $8.91839623$ by $\sqrt{6} + \sqrt{2}$ .

$c = \frac{34.45803701}{4}$

Step 9.4.3.3.6

Divide $34.45803701$ by $4$ .

$c = 8.61450925$

Step 10

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

Solve for the second triangle.

Step 11

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

Step 12

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

$\frac{\sin (B)}{5} = \frac{\sin (15)}{4}$

Step 13

Solve the equation for $B$ .

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Step 13.1

Multiply both sides of the equation by $5$ .

$5 \frac{\sin (B)}{5} = 5 \frac{\sin (15)}{4}$

Step 13.2

Simplify both sides of the equation.

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Step 13.2.1

Simplify the left side.

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Step 13.2.1.1

Cancel the common factor of $5$ .

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Step 13.2.1.1.1

Cancel the common factor.

$5 \frac{\sin (B)}{5} = 5 \frac{\sin (15)}{4}$

Step 13.2.1.1.2

Rewrite the expression.

$\sin (B) = 5 \frac{\sin (15)}{4}$

Step 13.2.2

Simplify the right side.

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Step 13.2.2.1

Simplify $5 \frac{\sin (15)}{4}$ .

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Step 13.2.2.1.1

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

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Step 13.2.2.1.1.1

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

$\sin (B) = 5 \frac{\sin (45 - 30)}{4}$

Step 13.2.2.1.1.2

Separate negation.

$\sin (B) = 5 \frac{\sin (45 - (30))}{4}$

Step 13.2.2.1.1.3

Apply the difference of angles identity.

$\sin (B) = 5 \frac{\sin (45) \cos (30) - \cos (45) \sin (30)}{4}$

Step 13.2.2.1.1.4

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

$\sin (B) = 5 \frac{\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30)}{4}$

Step 13.2.2.1.1.5

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

$\sin (B) = 5 \frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30)}{4}$

Step 13.2.2.1.1.6

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

$\sin (B) = 5 \frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30)}{4}$

Step 13.2.2.1.1.7

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

$\sin (B) = 5 \frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{4}$

Step 13.2.2.1.1.8

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

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Step 13.2.2.1.1.8.1

Simplify each term.

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Step 13.2.2.1.1.8.1.1

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

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Step 13.2.2.1.1.8.1.1.1

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

$\sin (B) = 5 \frac{\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{4}$

Step 13.2.2.1.1.8.1.1.2

Combine using the product rule for radicals.

$\sin (B) = 5 \frac{\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{4}$

Step 13.2.2.1.1.8.1.1.3

Multiply $2$ by $3$ .

$\sin (B) = 5 \frac{\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{4}$

Step 13.2.2.1.1.8.1.1.4

Multiply $2$ by $2$ .

$\sin (B) = 5 \frac{\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{4}$

Step 13.2.2.1.1.8.1.2

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

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Step 13.2.2.1.1.8.1.2.1

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

$\sin (B) = 5 \frac{\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}}{4}$

Step 13.2.2.1.1.8.1.2.2

Multiply $2$ by $2$ .

$\sin (B) = 5 \frac{\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}}{4}$

Step 13.2.2.1.1.8.2

Combine the numerators over the common denominator.

$\sin (B) = 5 \frac{\frac{\sqrt{6} - \sqrt{2}}{4}}{4}$

Step 13.2.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$\sin (B) = 5 (\frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{1}{4})$

Step 13.2.2.1.3

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

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Step 13.2.2.1.3.1

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

$\sin (B) = 5 \frac{\sqrt{6} - \sqrt{2}}{4 \cdot 4}$

Step 13.2.2.1.3.2

Multiply $4$ by $4$ .

$\sin (B) = 5 \frac{\sqrt{6} - \sqrt{2}}{16}$

Step 13.2.2.1.4

Combine $5$ and $\frac{\sqrt{6} - \sqrt{2}}{16}$ .

$\sin (B) = \frac{5 (\sqrt{6} - \sqrt{2})}{16}$

Step 13.3

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

$B = \arcsin (\frac{5 (\sqrt{6} - \sqrt{2})}{16})$

Step 13.4

Simplify the right side.

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Step 13.4.1

Evaluate $\arcsin (\frac{5 (\sqrt{6} - \sqrt{2})}{16})$ .

$B = 18.87616421$

Step 13.5

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.

$B = 180 - 18.87616421$

Step 13.6

Subtract $18.87616421$ from $180$ .

$B = 161.12383578$

Step 13.7

The solution to the equation $B = 18.87616421$ .

$B = 18.87616421, 161.12383578$

Step 14

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

$15 + C + 161.12383578 = 180$

Step 15

Solve the equation for $C$ .

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Step 15.1

Add $15$ and $161.12383578$ .

$C + 176.12383578 = 180$

Step 15.2

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

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Step 15.2.1

Subtract $176.12383578$ from both sides of the equation.

$C = 180 - 176.12383578$

Step 15.2.2

Subtract $176.12383578$ from $180$ .

$C = 3.87616421$

Step 16

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

Step 17

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

$\frac{\sin (3.87616421)}{c} = \frac{\sin (15)}{4}$

Step 18

Solve the equation for $c$ .

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Step 18.1

Factor each term.

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Step 18.1.1

Evaluate $\sin (3.87616421)$ .

$\frac{0.06760023}{c} = \frac{\sin (15)}{4}$

Step 18.1.2

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

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Step 18.1.2.1

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

$\frac{0.06760023}{c} = \frac{\sin (45 - 30)}{4}$

Step 18.1.2.2

Separate negation.

$\frac{0.06760023}{c} = \frac{\sin (45 - (30))}{4}$

Step 18.1.2.3

Apply the difference of angles identity.

$\frac{0.06760023}{c} = \frac{\sin (45) \cos (30) - \cos (45) \sin (30)}{4}$

Step 18.1.2.4

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

$\frac{0.06760023}{c} = \frac{\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30)}{4}$

Step 18.1.2.5

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

$\frac{0.06760023}{c} = \frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30)}{4}$

Step 18.1.2.6

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

$\frac{0.06760023}{c} = \frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30)}{4}$

Step 18.1.2.7

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

$\frac{0.06760023}{c} = \frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{4}$

Step 18.1.2.8

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

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Step 18.1.2.8.1

Simplify each term.

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Step 18.1.2.8.1.1

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

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Step 18.1.2.8.1.1.1

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

$\frac{0.06760023}{c} = \frac{\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{4}$

Step 18.1.2.8.1.1.2

Combine using the product rule for radicals.

$\frac{0.06760023}{c} = \frac{\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{4}$

Step 18.1.2.8.1.1.3

Multiply $2$ by $3$ .

$\frac{0.06760023}{c} = \frac{\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{4}$

Step 18.1.2.8.1.1.4

Multiply $2$ by $2$ .

$\frac{0.06760023}{c} = \frac{\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{4}$

Step 18.1.2.8.1.2

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

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Step 18.1.2.8.1.2.1

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

$\frac{0.06760023}{c} = \frac{\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}}{4}$

Step 18.1.2.8.1.2.2

Multiply $2$ by $2$ .

$\frac{0.06760023}{c} = \frac{\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}}{4}$

Step 18.1.2.8.2

Combine the numerators over the common denominator.

$\frac{0.06760023}{c} = \frac{\frac{\sqrt{6} - \sqrt{2}}{4}}{4}$

Step 18.1.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{0.06760023}{c} = \frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{1}{4}$

Step 18.1.4

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

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Step 18.1.4.1

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

$\frac{0.06760023}{c} = \frac{\sqrt{6} - \sqrt{2}}{4 \cdot 4}$

Step 18.1.4.2

Multiply $4$ by $4$ .

$\frac{0.06760023}{c} = \frac{\sqrt{6} - \sqrt{2}}{16}$

Step 18.2

Find the LCD of the terms in the equation.

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Step 18.2.1

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

$c, 16$

Step 18.2.2

Since $c, 16$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 16$ then find LCM for the variable part $c^{1}$ .

Step 18.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 18.2.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 18.2.5

The prime factors for $16$ are $2 \cdot 2 \cdot 2 \cdot 2$ .

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Step 18.2.5.1

$16$ has factors of $2$ and $8$ .

$2 \cdot 8$

Step 18.2.5.2

$8$ has factors of $2$ and $4$ .

$2 \cdot 2 \cdot 4$

Step 18.2.5.3

$4$ has factors of $2$ and $2$ .

$2 \cdot 2 \cdot 2 \cdot 2$

Step 18.2.6

Multiply $2 \cdot 2 \cdot 2 \cdot 2$ .

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Step 18.2.6.1

Multiply $2$ by $2$ .

$4 \cdot 2 \cdot 2$

Step 18.2.6.2

Multiply $4$ by $2$ .

$8 \cdot 2$

Step 18.2.6.3

Multiply $8$ by $2$ .

$16$

Step 18.2.7

The factor for $c^{1}$ is $c$ itself.

$c^{1} = c$

$c$ occurs $1$ time.

Step 18.2.8

The LCM of $c^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$c$

Step 18.2.9

The LCM for $c, 16$ is the numeric part $16$ multiplied by the variable part.

$16 c$

Step 18.3

Multiply each term in $\frac{0.06760023}{c} = \frac{\sqrt{6} - \sqrt{2}}{16}$ by $16 c$ to eliminate the fractions.

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Step 18.3.1

Multiply each term in $\frac{0.06760023}{c} = \frac{\sqrt{6} - \sqrt{2}}{16}$ by $16 c$ .

$\frac{0.06760023}{c} (16 c) = \frac{\sqrt{6} - \sqrt{2}}{16} (16 c)$

Step 18.3.2

Simplify the left side.

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Step 18.3.2.1

Rewrite using the commutative property of multiplication.

$16 \frac{0.06760023}{c} c = \frac{\sqrt{6} - \sqrt{2}}{16} (16 c)$

Step 18.3.2.2

Multiply $16 \frac{0.06760023}{c}$ .

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Step 18.3.2.2.1

Combine $16$ and $\frac{0.06760023}{c}$ .

$\frac{16 \cdot 0.06760023}{c} c = \frac{\sqrt{6} - \sqrt{2}}{16} (16 c)$

Step 18.3.2.2.2

Multiply $16$ by $0.06760023$ .

$\frac{1.08160376}{c} c = \frac{\sqrt{6} - \sqrt{2}}{16} (16 c)$

Step 18.3.2.3

Cancel the common factor of $c$ .

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Step 18.3.2.3.1

Cancel the common factor.

$\frac{1.08160376}{c} c = \frac{\sqrt{6} - \sqrt{2}}{16} (16 c)$

Step 18.3.2.3.2

Rewrite the expression.

$1.08160376 = \frac{\sqrt{6} - \sqrt{2}}{16} (16 c)$

Step 18.3.3

Simplify the right side.

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Step 18.3.3.1

Cancel the common factor of $16$ .

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Step 18.3.3.1.1

Factor $16$ out of $16 c$ .

$1.08160376 = \frac{\sqrt{6} - \sqrt{2}}{16} (16 (c))$

Step 18.3.3.1.2

Cancel the common factor.

$1.08160376 = \frac{\sqrt{6} - \sqrt{2}}{16} (16 c)$

Step 18.3.3.1.3

Rewrite the expression.

$1.08160376 = (\sqrt{6} - \sqrt{2}) c$

Step 18.3.3.2

Apply the distributive property.

$1.08160376 = \sqrt{6} c - \sqrt{2} c$

Step 18.4

Solve the equation.

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Step 18.4.1

Rewrite the equation as $\sqrt{6} c - \sqrt{2} c = 1.08160376$ .

$\sqrt{6} c - \sqrt{2} c = 1.08160376$

Step 18.4.2

Factor $c$ out of $\sqrt{6} c - \sqrt{2} c$ .

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Step 18.4.2.1

Factor $c$ out of $\sqrt{6} c$ .

$c \sqrt{6} - \sqrt{2} c = 1.08160376$

Step 18.4.2.2

Factor $c$ out of $- \sqrt{2} c$ .

$c \sqrt{6} + c (- \sqrt{2}) = 1.08160376$

Step 18.4.2.3

Factor $c$ out of $c \sqrt{6} + c (- \sqrt{2})$ .

$c (\sqrt{6} - \sqrt{2}) = 1.08160376$

Step 18.4.3

Divide each term in $c (\sqrt{6} - \sqrt{2}) = 1.08160376$ by $\sqrt{6} - \sqrt{2}$ and simplify.

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Step 18.4.3.1

Divide each term in $c (\sqrt{6} - \sqrt{2}) = 1.08160376$ by $\sqrt{6} - \sqrt{2}$ .

$\frac{c (\sqrt{6} - \sqrt{2})}{\sqrt{6} - \sqrt{2}} = \frac{1.08160376}{\sqrt{6} - \sqrt{2}}$

Step 18.4.3.2

Simplify the left side.

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Step 18.4.3.2.1

Cancel the common factor of $\sqrt{6} - \sqrt{2}$ .

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Step 18.4.3.2.1.1

Cancel the common factor.

$\frac{c (\sqrt{6} - \sqrt{2})}{\sqrt{6} - \sqrt{2}} = \frac{1.08160376}{\sqrt{6} - \sqrt{2}}$

Step 18.4.3.2.1.2

Divide $c$ by $1$ .

$c = \frac{1.08160376}{\sqrt{6} - \sqrt{2}}$

Step 18.4.3.3

Simplify the right side.

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Step 18.4.3.3.1

Multiply $\frac{1.08160376}{\sqrt{6} - \sqrt{2}}$ by $\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}$ .

$c = \frac{1.08160376}{\sqrt{6} - \sqrt{2}} \cdot \frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}$

Step 18.4.3.3.2

Multiply $\frac{1.08160376}{\sqrt{6} - \sqrt{2}}$ by $\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}$ .

$c = \frac{1.08160376 (\sqrt{6} + \sqrt{2})}{(\sqrt{6} - \sqrt{2}) (\sqrt{6} + \sqrt{2})}$

Step 18.4.3.3.3

Expand the denominator using the FOIL method.

$c = \frac{1.08160376 (\sqrt{6} + \sqrt{2})}{{\sqrt{6}}^{2} + \sqrt{12} - \sqrt{12} - {\sqrt{2}}^{2}}$

Step 18.4.3.3.4

Simplify.

$c = \frac{1.08160376 (\sqrt{6} + \sqrt{2})}{4}$

Step 18.4.3.3.5

Multiply $1.08160376$ by $\sqrt{6} + \sqrt{2}$ .

$c = \frac{4.17899603}{4}$

Step 18.4.3.3.6

Divide $4.17899603$ by $4$ .

$c = 1.044749$

Step 19

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

First Triangle Combination:

$A = 15$

$B = 18.87616421$

$C = 146.12383578$

$a = 4$

$b = 5$

$c = 8.61450925$

Second Triangle Combination:

$A = 15$

$B = 161.12383578$

$C = 3.87616421$

$a = 4$

$b = 5$

$c = 1.044749$