Trigonometry Examples

a = 3

$a = 3$ ,

b = 4

$b = 4$ ,

C = 52

$C = 52$

Step 1

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

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

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

Step 2

Solve the equation.

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

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

Step 3

Substitute the known values into the equation.

c = \sqrt{{(3)}^{2} + {(4)}^{2} - 2 \cdot 3 \cdot 4 \cos (52)}

$c = \sqrt{{(3)}^{2} + {(4)}^{2} - 2 \cdot 3 \cdot 4 \cos (52)}$

Step 4

Simplify the results.

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Step 4.1

Raise

3

$3$ to the power of

2

$2$ .

c = \sqrt{9 + {(4)}^{2} - 2 \cdot 3 \cdot (4 \cos (52))}

$c = \sqrt{9 + {(4)}^{2} - 2 \cdot 3 \cdot (4 \cos (52))}$

Step 4.2

Raise

4

$4$ to the power of

2

$2$ .

c = \sqrt{9 + 16 - 2 \cdot 3 \cdot (4 \cos (52))}

$c = \sqrt{9 + 16 - 2 \cdot 3 \cdot (4 \cos (52))}$

Step 4.3

Multiply

- 2 \cdot 3 \cdot 4

$- 2 \cdot 3 \cdot 4$ .

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Step 4.3.1

Multiply

- 2

$- 2$ by

3

$3$ .

c = \sqrt{9 + 16 - 6 \cdot (4 \cos (52))}

$c = \sqrt{9 + 16 - 6 \cdot (4 \cos (52))}$

Step 4.3.2

Multiply

- 6

$- 6$ by

4

$4$ .

c = \sqrt{9 + 16 - 24 \cos (52)}

$c = \sqrt{9 + 16 - 24 \cos (52)}$

c = \sqrt{9 + 16 - 24 \cos (52)}

$c = \sqrt{9 + 16 - 24 \cos (52)}$

Step 4.4

Evaluate

\cos (52)

$\cos (52)$ .

c = \sqrt{9 + 16 - 24 \cdot 0.61566147}

$c = \sqrt{9 + 16 - 24 \cdot 0.61566147}$

Step 4.5

Multiply

- 24

$- 24$ by

0.61566147

$0.61566147$ .

c = \sqrt{9 + 16 - 14.7758754}

$c = \sqrt{9 + 16 - 14.7758754}$

Step 4.6

Add

9

$9$ and

16

$16$ .

c = \sqrt{25 - 14.7758754}

$c = \sqrt{25 - 14.7758754}$

Step 4.7

Subtract

14.7758754

$14.7758754$ from

25

$25$ .

c = \sqrt{10.22412459}

$c = \sqrt{10.22412459}$

Step 4.8

Evaluate the root.

c = 3.1975185

$c = 3.1975185$

c = 3.1975185

$c = 3.1975185$

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}

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

Step 6

Substitute the known values into the law of sines to find

A

$A$ .

\frac{\sin (A)}{3} = \frac{\sin (52)}{3.1975185}

$\frac{\sin (A)}{3} = \frac{\sin (52)}{3.1975185}$

Step 7

Solve the equation for

A

$A$ .

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Step 7.1

Multiply both sides of the equation by

3

$3$ .

3 \frac{\sin (A)}{3} = 3 \frac{\sin (52)}{3.1975185}

$3 \frac{\sin (A)}{3} = 3 \frac{\sin (52)}{3.1975185}$

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

3

$3$ .

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Step 7.2.1.1.1

Cancel the common factor.

3 \frac{\sin (A)}{3} = 3 \frac{\sin (52)}{3.1975185}

$3 \frac{\sin (A)}{3} = 3 \frac{\sin (52)}{3.1975185}$

Step 7.2.1.1.2

Rewrite the expression.

\sin (A) = 3 \frac{\sin (52)}{3.1975185}

$\sin (A) = 3 \frac{\sin (52)}{3.1975185}$

\sin (A) = 3 \frac{\sin (52)}{3.1975185}

$\sin (A) = 3 \frac{\sin (52)}{3.1975185}$

\sin (A) = 3 \frac{\sin (52)}{3.1975185}

$\sin (A) = 3 \frac{\sin (52)}{3.1975185}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify

3 \frac{\sin (52)}{3.1975185}

$3 \frac{\sin (52)}{3.1975185}$ .

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Step 7.2.2.1.1

Evaluate

\sin (52)

$\sin (52)$ .

\sin (A) = 3 (\frac{0.78801075}{3.1975185})

$\sin (A) = 3 (\frac{0.78801075}{3.1975185})$

Step 7.2.2.1.2

Divide

0.78801075

$0.78801075$ by

3.1975185

$3.1975185$ .

\sin (A) = 3 \cdot 0.24644447

$\sin (A) = 3 \cdot 0.24644447$

Step 7.2.2.1.3

Multiply

3

$3$ by

0.24644447

$0.24644447$ .

\sin (A) = 0.73933341

$\sin (A) = 0.73933341$

\sin (A) = 0.73933341

$\sin (A) = 0.73933341$

\sin (A) = 0.73933341

$\sin (A) = 0.73933341$

\sin (A) = 0.73933341

$\sin (A) = 0.73933341$

Step 7.3

Take the inverse sine of both sides of the equation to extract

A

$A$ from inside the sine.

A = \arcsin (0.73933341)

$A = \arcsin (0.73933341)$

Step 7.4

Simplify the right side.

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Step 7.4.1

Evaluate

\arcsin (0.73933341)

$\arcsin (0.73933341)$ .

A = 47.67466326

$A = 47.67466326$

A = 47.67466326

$A = 47.67466326$

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

$180$ to find the solution in the second quadrant.

A = 180 - 47.67466326

$A = 180 - 47.67466326$

Step 7.6

Subtract

47.67466326

$47.67466326$ from

180

$180$ .

A = 132.32533673

$A = 132.32533673$

Step 7.7

The solution to the equation

A = 47.67466326

$A = 47.67466326$ .

A = 47.67466326, 132.32533673

$A = 47.67466326, 132.32533673$

Step 7.8

Exclude the invalid angle.

A = 47.67466326

$A = 47.67466326$

A = 47.67466326

$A = 47.67466326$

Step 8

The sum of all the angles in a triangle is

180

$180$ degrees.

47.67466326 + 52 + B = 180

$47.67466326 + 52 + B = 180$

Step 9

Solve the equation for

B

$B$ .

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Step 9.1

Add

47.67466326

$47.67466326$ and

52

$52$ .

99.67466326 + B = 180

$99.67466326 + B = 180$

Step 9.2

Move all terms not containing

B

$B$ to the right side of the equation.

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Step 9.2.1

Subtract

99.67466326

$99.67466326$ from both sides of the equation.

B = 180 - 99.67466326

$B = 180 - 99.67466326$

Step 9.2.2

Subtract

99.67466326

$99.67466326$ from

180

$180$ .

B = 80.32533673

$B = 80.32533673$

B = 80.32533673

$B = 80.32533673$

B = 80.32533673

$B = 80.32533673$

Step 10

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

A = 47.67466326

$A = 47.67466326$

B = 80.32533673

$B = 80.32533673$

C = 52

$C = 52$

a = 3

$a = 3$

b = 4

$b = 4$

c = 3.1975185

$c = 3.1975185$