Trigonometry Examples

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\begin{matrix} Side & Angle \\ \begin{matrix} b = & 5.6 \\ c = & 10.6 \\ a = \end{matrix} & \begin{matrix} A = \\ B = \\ C = \end{matrix} \end{matrix}

$\begin{matrix} Side & Angle \\ \begin{matrix} b = & 5.6 \\ c = & 10.6 \\ a = \end{matrix} & \begin{matrix} A = \\ B = \\ C = \end{matrix} \end{matrix}$

Step 1

Assume that angle

C = 90

$C = 90$ .

C = 90

$C = 90$

Step 2

Find the last side of the triangle using the Pythagorean theorem.

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Step 2.1

Use the Pythagorean theorem to find the unknown side. In any right triangle, the area of the square whose side is the hypotenuse (the side of a right triangle opposite the right angle) is equal to the sum of areas of the squares whose sides are the two legs (the two sides other than the hypotenuse).

a^{2} + b^{2} = c^{2}

$a^{2} + b^{2} = c^{2}$

Step 2.2

Solve the equation for

a

$a$ .

a = \sqrt{c^{2} - b^{2}}

$a = \sqrt{c^{2} - b^{2}}$

Step 2.3

Substitute the actual values into the equation.

a = \sqrt{{(10.6)}^{2} - {(5.6)}^{2}}

$a = \sqrt{{(10.6)}^{2} - {(5.6)}^{2}}$

Step 2.4

Raise

10.6

$10.6$ to the power of

2

$2$ .

a = \sqrt{112.36 - {(5.6)}^{2}}

$a = \sqrt{112.36 - {(5.6)}^{2}}$

Step 2.5

Raise

5.6

$5.6$ to the power of

2

$2$ .

a = \sqrt{112.36 - 1 \cdot 31.36}

$a = \sqrt{112.36 - 1 \cdot 31.36}$

Step 2.6

Multiply

- 1

$- 1$ by

31.36

$31.36$ .

a = \sqrt{112.36 - 31.36}

$a = \sqrt{112.36 - 31.36}$

Step 2.7

Subtract

31.36

$31.36$ from

112.36

$112.36$ .

a = \sqrt{81}

$a = \sqrt{81}$

Step 2.8

Rewrite

81

$81$ as

9^{2}

$9^{2}$ .

a = \sqrt{9^{2}}

$a = \sqrt{9^{2}}$

Step 2.9

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

a = 9

$a = 9$

a = 9

$a = 9$

Step 3

Find

B

$B$ .

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Step 3.1

The angle

B

$B$ can be found using the inverse sine function.

B = \arcsin (\frac{opp}{hyp})

$B = \arcsin (\frac{opp}{hyp})$

Step 3.2

Substitute in the values of the opposite side to angle

B

$B$ and hypotenuse

10.6

$10.6$ of the triangle.

B = \arcsin (\frac{5.6}{10.6})

$B = \arcsin (\frac{5.6}{10.6})$

Step 3.3

Divide

5.6

$5.6$ by

10.6

$10.6$ .

B = \arcsin (0.52830188)

$B = \arcsin (0.52830188)$

Step 3.4

Evaluate

\arcsin (0.52830188)

$\arcsin (0.52830188)$ .

B = 31.8907918

$B = 31.8907918$

B = 31.8907918

$B = 31.8907918$

Step 4

Find the last angle of the triangle.

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Step 4.1

The sum of all the angles in a triangle is

180

$180$ degrees.

A + 90 + 31.8907918 = 180

$A + 90 + 31.8907918 = 180$

Step 4.2

Solve the equation for

A

$A$ .

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Step 4.2.1

Add

90

$90$ and

31.8907918

$31.8907918$ .

A + 121.8907918 = 180

$A + 121.8907918 = 180$

Step 4.2.2

Move all terms not containing

A

$A$ to the right side of the equation.

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Step 4.2.2.1

Subtract

121.8907918

$121.8907918$ from both sides of the equation.

A = 180 - 121.8907918

$A = 180 - 121.8907918$

Step 4.2.2.2

Subtract

121.8907918

$121.8907918$ from

180

$180$ .

A = 58.10920819

$A = 58.10920819$

A = 58.10920819

$A = 58.10920819$

A = 58.10920819

$A = 58.10920819$

A = 58.10920819

$A = 58.10920819$

Step 5

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

A = 58.10920819

$A = 58.10920819$

B = 31.8907918

$B = 31.8907918$

C = 90

$C = 90$

a = 9

$a = 9$

b = 5.6

$b = 5.6$

c = 10.6

$c = 10.6$