Trigonometry Examples

$a = 8$ , $c = 13$ , $B = 150$

Step 1

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

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

Step 2

Solve the equation.

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

Step 3

Substitute the known values into the equation.

$b = \sqrt{{(8)}^{2} + {(13)}^{2} - 2 \cdot 8 \cdot 13 \cos (150)}$

Step 4

Simplify the results.

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Step 4.1

Raise $8$ to the power of $2$ .

$b = \sqrt{64 + {(13)}^{2} - 2 \cdot 8 \cdot (13 \cos (150))}$

Step 4.2

Raise $13$ to the power of $2$ .

$b = \sqrt{64 + 169 - 2 \cdot 8 \cdot (13 \cos (150))}$

Step 4.3

Multiply $- 2 \cdot 8 \cdot 13$ .

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Step 4.3.1

Multiply $- 2$ by $8$ .

$b = \sqrt{64 + 169 - 16 \cdot (13 \cos (150))}$

Step 4.3.2

Multiply $- 16$ by $13$ .

$b = \sqrt{64 + 169 - 208 \cos (150)}$

Step 4.4

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$b = \sqrt{64 + 169 - 208 (- \cos (30))}$

Step 4.5

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

$b = \sqrt{64 + 169 - 208 (- \frac{\sqrt{3}}{2})}$

Step 4.6

Cancel the common factor of $2$ .

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Step 4.6.1

Move the leading negative in $- \frac{\sqrt{3}}{2}$ into the numerator.

$b = \sqrt{64 + 169 - 208 \frac{- \sqrt{3}}{2}}$

Step 4.6.2

Factor $2$ out of $- 208$ .

$b = \sqrt{64 + 169 + 2 (- 104) (\frac{- \sqrt{3}}{2})}$

Step 4.6.3

Cancel the common factor.

$b = \sqrt{64 + 169 + 2 \cdot (- 104 \frac{- \sqrt{3}}{2})}$

Step 4.6.4

Rewrite the expression.

$b = \sqrt{64 + 169 - 104 (- \sqrt{3})}$

Step 4.7

Simplify the expression.

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Step 4.7.1

Multiply $- 1$ by $- 104$ .

$b = \sqrt{64 + 169 + 104 \sqrt{3}}$

Step 4.7.2

Add $64$ and $169$ .

$b = \sqrt{233 + 104 \sqrt{3}}$

Step 5

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 6

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

$\frac{\sin (A)}{8} = \frac{\sin (150)}{\sqrt{233 + 104 \sqrt{3}}}$

Step 7

Solve the equation for $A$ .

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Step 7.1

Multiply both sides of the equation by $8$ .

$8 \frac{\sin (A)}{8} = 8 \frac{\sin (150)}{\sqrt{233 + 104 \sqrt{3}}}$

Step 7.2

Simplify both sides of the equation.

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Step 7.2.1

Simplify the left side.

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Step 7.2.1.1

Cancel the common factor of $8$ .

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Step 7.2.1.1.1

Cancel the common factor.

$8 \frac{\sin (A)}{8} = 8 \frac{\sin (150)}{\sqrt{233 + 104 \sqrt{3}}}$

Step 7.2.1.1.2

Rewrite the expression.

$\sin (A) = 8 \frac{\sin (150)}{\sqrt{233 + 104 \sqrt{3}}}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $8 \frac{\sin (150)}{\sqrt{233 + 104 \sqrt{3}}}$ .

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Step 7.2.2.1.1

Simplify the numerator.

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Step 7.2.2.1.1.1

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

$\sin (A) = 8 \frac{\sin (30)}{\sqrt{233 + 104 \sqrt{3}}}$

Step 7.2.2.1.1.2

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

$\sin (A) = 8 \frac{\frac{1}{2}}{\sqrt{233 + 104 \sqrt{3}}}$

Step 7.2.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$\sin (A) = 8 (\frac{1}{2} \cdot \frac{1}{\sqrt{233 + 104 \sqrt{3}}})$

Step 7.2.2.1.3

Multiply $\frac{1}{2}$ by $\frac{1}{\sqrt{233 + 104 \sqrt{3}}}$ .

$\sin (A) = 8 \frac{1}{2 \sqrt{233 + 104 \sqrt{3}}}$

Step 7.2.2.1.4

Cancel the common factor of $2$ .

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Step 7.2.2.1.4.1

Factor $2$ out of $8$ .

$\sin (A) = 2 (4) \frac{1}{2 \sqrt{233 + 104 \sqrt{3}}}$

Step 7.2.2.1.4.2

Factor $2$ out of $2 \sqrt{233 + 104 \sqrt{3}}$ .

$\sin (A) = 2 (4) \frac{1}{2 (\sqrt{233 + 104 \sqrt{3}})}$

Step 7.2.2.1.4.3

Cancel the common factor.

$\sin (A) = 2 \cdot 4 \frac{1}{2 \sqrt{233 + 104 \sqrt{3}}}$

Step 7.2.2.1.4.4

Rewrite the expression.

$\sin (A) = 4 \frac{1}{\sqrt{233 + 104 \sqrt{3}}}$

Step 7.2.2.1.5

Combine $4$ and $\frac{1}{\sqrt{233 + 104 \sqrt{3}}}$ .

$\sin (A) = \frac{4}{\sqrt{233 + 104 \sqrt{3}}}$

Step 7.3

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

$A = \arcsin (\frac{4}{\sqrt{233 + 104 \sqrt{3}}})$

Step 7.4

Simplify the right side.

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Step 7.4.1

Evaluate $\arcsin (\frac{4}{\sqrt{233 + 104 \sqrt{3}}})$ .

$A = 11.34962381$

Step 7.5

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.

$A = 180 - 11.34962381$

Step 7.6

Subtract $11.34962381$ from $180$ .

$A = 168.65037618$

Step 7.7

The solution to the equation $A = 11.34962381$ .

$A = 11.34962381, 168.65037618$

Step 7.8

Exclude the invalid angle.

$A = 11.34962381$

Step 8

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

$11.34962381 + C + 150 = 180$

Step 9

Solve the equation for $C$ .

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Step 9.1

Add $11.34962381$ and $150$ .

$C + 161.34962381 = 180$

Step 9.2

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

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Step 9.2.1

Subtract $161.34962381$ from both sides of the equation.

$C = 180 - 161.34962381$

Step 9.2.2

Subtract $161.34962381$ from $180$ .

$C = 18.65037618$

Step 10

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

$A = 11.34962381$

$B = 150$

$C = 18.65037618$

$a = 8$

$b = \sqrt{233 + 104 \sqrt{3}}$

$c = 13$