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}

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

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}

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

Step 2

Substitute the known values into the law of sines to find

A

$A$ .

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

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

Step 3

Solve the equation for

A

$A$ .

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Step 3.1

Multiply both sides of the equation by

14

$14$ .

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

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

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

$14$ .

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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.

$A = 36.01201213$

$B = 105$

$C = 38.98798786$

$a = 14$

$b = 23$

$c = 14.98108954$