Trigonometry Examples

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\begin{matrix} Side & Angle \begin{matrix} b = & 4 c = a = & 3 \end{matrix} & \begin{matrix} A = B = C = & 90 ° \end{matrix} \end{matrix}

$\begin{matrix} Side & Angle \\ \begin{matrix} b = & 4 \\ c = \\ a = & 3 \end{matrix} & \begin{matrix} A = \\ B = \\ C = & 90 ° \end{matrix} \end{matrix}$

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 (90 °)}

$c = \sqrt{{(3)}^{2} + {(4)}^{2} - 2 \cdot 3 \cdot 4 \cos (90 °)}$

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 (90 °))}

$c = \sqrt{9 + {(4)}^{2} - 2 \cdot 3 \cdot (4 \cos (90 °))}$

Step 4.2

Raise

4

$4$ to the power of

2

$2$ .

c = \sqrt{9 + 16 - 2 \cdot 3 \cdot (4 cos (90 °))}

$c = \sqrt{9 + 16 - 2 \cdot 3 \cdot (4 \cos (90 °))}$

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 (90 °))}

$c = \sqrt{9 + 16 - 6 \cdot (4 \cos (90 °))}$

Step 4.3.2

Multiply

- 6

$- 6$ by

4

$4$ .

c = \sqrt{9 + 16 - 24 cos (90 °)}

$c = \sqrt{9 + 16 - 24 \cos (90 °)}$

c = \sqrt{9 + 16 - 24 cos (90 °)}

$c = \sqrt{9 + 16 - 24 \cos (90 °)}$

Step 4.4

The exact value of

cos (90 °)

$\cos (90 °)$ is

0

$0$ .

c = \sqrt{9 + 16 - 24 \cdot 0}

$c = \sqrt{9 + 16 - 24 \cdot 0}$

Step 4.5

Multiply

- 24

$- 24$ by

0

$0$ .

c = \sqrt{9 + 16 + 0}

$c = \sqrt{9 + 16 + 0}$

Step 4.6

Add

9 + 16

$9 + 16$ and

0

$0$ .

c = \sqrt{9 + 16}

$c = \sqrt{9 + 16}$

Step 4.7

Add

9

$9$ and

16

$16$ .

c = \sqrt{25}

$c = \sqrt{25}$

Step 4.8

Rewrite

25

$25$ as

5^{2}

$5^{2}$ .

c = \sqrt{5^{2}}

$c = \sqrt{5^{2}}$

Step 4.9

Pull terms out from under the radical, assuming positive real numbers.

c = 5

$c = 5$

c = 5

$c = 5$

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 (90 °)}{5}

$\frac{\sin (A)}{3} = \frac{\sin (90 °)}{5}$

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 (90 °)}{5}

$3 \frac{\sin (A)}{3} = 3 \frac{\sin (90 °)}{5}$

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.

$3 \frac{\sin (A)}{3} = 3 \frac{\sin (90 °)}{5}$

Step 7.2.1.1.2

Rewrite the expression.

$\sin (A) = 3 \frac{\sin (90 °)}{5}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify

$3 \frac{\sin (90 °)}{5}$ .

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Step 7.2.2.1.1

The exact value of

$\sin (90 °)$ is

$1$ .

$\sin (A) = 3 (\frac{1}{5})$

Step 7.2.2.1.2

Combine

$3$ and

$\frac{1}{5}$ .

$\sin (A) = \frac{3}{5}$

Step 7.3

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

$A$ from inside the sine.

$A = \arcsin (\frac{3}{5})$

Step 7.4

Simplify the right side.

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Step 7.4.1

Evaluate

$\arcsin (\frac{3}{5})$ .

$A = 36.86989764$

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.

$A = 180 - 36.86989764$

Step 7.6

Subtract

$36.86989764$ from

$180$ .

$A = 143.13010235$

Step 7.7

The solution to the equation

$A = 36.86989764$ .

$A = 36.86989764, 143.13010235$

Step 7.8

Exclude the invalid angle.

$A = 36.86989764$

Step 8

The sum of all the angles in a triangle is

$180$ degrees.

$36.86989764 + 90 ° + B = 180$

Step 9

Solve the equation for

$B$ .

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Step 9.1

Add

$36.86989764$ and

$90 °$ .

$126.86989764 + B = 180$

Step 9.2

Move all terms not containing

$B$ to the right side of the equation.

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Step 9.2.1

Subtract

$126.86989764$ from both sides of the equation.

$B = 180 - 126.86989764$

Step 9.2.2

Subtract

$126.86989764$ from

$180$ .

$B = 53.13010235$

Step 10

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

$A = 36.86989764$

$B = 53.13010235$

$C = 90 °$

$a = 3$

$b = 4$

$c = 5$