Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 23 \\ c = \\ a = & 14 \end{matrix} & \begin{matrix} A = \\ B = & 105 \\ C = \end{matrix} \end{matrix}$

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 $A$ .

$\frac{\sin (A)}{14} = \frac{\sin (105)}{23}$

Step 3

Solve the equation for $A$ .

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Step 3.1

Multiply both sides of the equation by $14$ .

$14 \frac{\sin (A)}{14} = 14 \frac{\sin (105)}{23}$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Cancel the common factor of $14$ .

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Step 3.2.1.1.1

Cancel the common factor.

$14 \frac{\sin (A)}{14} = 14 \frac{\sin (105)}{23}$

Step 3.2.1.1.2

Rewrite the expression.

$\sin (A) = 14 \frac{\sin (105)}{23}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $14 \frac{\sin (105)}{23}$ .

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Step 3.2.2.1.1

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

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Step 3.2.2.1.1.1

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

$\sin (A) = 14 \frac{\sin (75)}{23}$

Step 3.2.2.1.1.2

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

$\sin (A) = 14 \frac{\sin (30 + 45)}{23}$

Step 3.2.2.1.1.3

Apply the sum of angles identity.

$\sin (A) = 14 \frac{\sin (30) \cos (45) + \cos (30) \sin (45)}{23}$

Step 3.2.2.1.1.4

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

$\sin (A) = 14 \frac{\frac{1}{2} \cos (45) + \cos (30) \sin (45)}{23}$

Step 3.2.2.1.1.5

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

$\sin (A) = 14 \frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45)}{23}$

Step 3.2.2.1.1.6

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

$\sin (A) = 14 \frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \sin (45)}{23}$

Step 3.2.2.1.1.7

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

$\sin (A) = 14 \frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{23}$

Step 3.2.2.1.1.8

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

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Step 3.2.2.1.1.8.1

Simplify each term.

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Step 3.2.2.1.1.8.1.1

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

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Step 3.2.2.1.1.8.1.1.1

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

$\sin (A) = 14 \frac{\frac{\sqrt{2}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{23}$

Step 3.2.2.1.1.8.1.1.2

Multiply $2$ by $2$ .

$\sin (A) = 14 \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{23}$

Step 3.2.2.1.1.8.1.2

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

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Step 3.2.2.1.1.8.1.2.1

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

$\sin (A) = 14 \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}}{23}$

Step 3.2.2.1.1.8.1.2.2

Combine using the product rule for radicals.

$\sin (A) = 14 \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}}{23}$

Step 3.2.2.1.1.8.1.2.3

Multiply $3$ by $2$ .

$\sin (A) = 14 \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{2 \cdot 2}}{23}$

Step 3.2.2.1.1.8.1.2.4

Multiply $2$ by $2$ .

$\sin (A) = 14 \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}{23}$

Step 3.2.2.1.1.8.2

Combine the numerators over the common denominator.

$\sin (A) = 14 \frac{\frac{\sqrt{2} + \sqrt{6}}{4}}{23}$

Step 3.2.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$\sin (A) = 14 (\frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{1}{23})$

Step 3.2.2.1.3

Multiply $\frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{1}{23}$ .

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Step 3.2.2.1.3.1

Multiply $\frac{\sqrt{2} + \sqrt{6}}{4}$ by $\frac{1}{23}$ .

$\sin (A) = 14 \frac{\sqrt{2} + \sqrt{6}}{4 \cdot 23}$

Step 3.2.2.1.3.2

Multiply $4$ by $23$ .

$\sin (A) = 14 \frac{\sqrt{2} + \sqrt{6}}{92}$

Step 3.2.2.1.4

Cancel the common factor of $2$ .

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Step 3.2.2.1.4.1

Factor $2$ out of $14$ .

$\sin (A) = 2 (7) \frac{\sqrt{2} + \sqrt{6}}{92}$

Step 3.2.2.1.4.2

Factor $2$ out of $92$ .

$\sin (A) = 2 \cdot 7 \frac{\sqrt{2} + \sqrt{6}}{2 \cdot 46}$

Step 3.2.2.1.4.3

Cancel the common factor.

$\sin (A) = 2 \cdot 7 \frac{\sqrt{2} + \sqrt{6}}{2 \cdot 46}$

Step 3.2.2.1.4.4

Rewrite the expression.

$\sin (A) = 7 \frac{\sqrt{2} + \sqrt{6}}{46}$

Step 3.2.2.1.5

Combine $7$ and $\frac{\sqrt{2} + \sqrt{6}}{46}$ .

$\sin (A) = \frac{7 (\sqrt{2} + \sqrt{6})}{46}$

Step 3.3

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

$A = \arcsin (\frac{7 (\sqrt{2} + \sqrt{6})}{46})$

Step 3.4

Simplify the right side.

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Step 3.4.1

Evaluate $\arcsin (\frac{7 (\sqrt{2} + \sqrt{6})}{46})$ .

$A = 36.01201213$

Step 3.5

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.

$A = 180 - 36.01201213$

Step 3.6

Subtract $36.01201213$ from $180$ .

$A = 143.98798786$

Step 3.7

The solution to the equation $A = 36.01201213$ .

$A = 36.01201213, 143.98798786$

Step 3.8

Exclude the invalid angle.

$A = 36.01201213$

Step 4

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

$36.01201213 + C + 105 = 180$

Step 5

Solve the equation for $C$ .

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Step 5.1

Add $36.01201213$ and $105$ .

$C + 141.01201213 = 180$

Step 5.2

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

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Step 5.2.1

Subtract $141.01201213$ from both sides of the equation.

$C = 180 - 141.01201213$

Step 5.2.2

Subtract $141.01201213$ from $180$ .

$C = 38.98798786$

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 $c$ .

$\frac{\sin (38.98798786)}{c} = \frac{\sin (36.01201213)}{14}$

Step 8

Solve the equation for $c$ .

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Step 8.1

Factor each term.

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Step 8.1.1

Evaluate $\sin (38.98798786)$ .

$\frac{0.62915744}{c} = \frac{\sin (36.01201213)}{14}$

Step 8.1.2

Evaluate $\sin (36.01201213)$ .

$\frac{0.62915744}{c} = \frac{0.58795485}{14}$

Step 8.1.3

Divide $0.58795485$ by $14$ .

$\frac{0.62915744}{c} = 0.04199677$

Step 8.2

Find the LCD of the terms in the equation.

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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.

$c, 1$

Step 8.2.2

The LCM of one and any expression is the expression.

$c$

Step 8.3

Multiply each term in $\frac{0.62915744}{c} = 0.04199677$ by $c$ to eliminate the fractions.

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Step 8.3.1

Multiply each term in $\frac{0.62915744}{c} = 0.04199677$ by $c$ .

$\frac{0.62915744}{c} c = 0.04199677 c$

Step 8.3.2

Simplify the left side.

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Step 8.3.2.1

Cancel the common factor of $c$ .

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Step 8.3.2.1.1

Cancel the common factor.

$\frac{0.62915744}{c} c = 0.04199677 c$

Step 8.3.2.1.2

Rewrite the expression.

$0.62915744 = 0.04199677 c$

Step 8.4

Solve the equation.

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Step 8.4.1

Rewrite the equation as $0.04199677 c = 0.62915744$ .

$0.04199677 c = 0.62915744$

Step 8.4.2

Divide each term in $0.04199677 c = 0.62915744$ by $0.04199677$ and simplify.

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Step 8.4.2.1

Divide each term in $0.04199677 c = 0.62915744$ by $0.04199677$ .

$\frac{0.04199677 c}{0.04199677} = \frac{0.62915744}{0.04199677}$

Step 8.4.2.2

Simplify the left side.

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Step 8.4.2.2.1

Cancel the common factor of $0.04199677$ .

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Step 8.4.2.2.1.1

Cancel the common factor.

$\frac{0.04199677 c}{0.04199677} = \frac{0.62915744}{0.04199677}$

Step 8.4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{0.62915744}{0.04199677}$

Step 8.4.2.3

Simplify the right side.

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Step 8.4.2.3.1

Divide $0.62915744$ by $0.04199677$ .

$c = 14.98108954$

Step 9

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

$A = 36.01201213$

$B = 105$

$C = 38.98798786$

$a = 14$

$b = 23$

$c = 14.98108954$