Trigonometry Examples



\begin{matrix} Side & Angle \\ \begin{matrix} b = & 5 \\ c = & 13 \\ a = \end{matrix} & \begin{matrix} A = \\ B = \\ C = \end{matrix} \end{matrix}

$\begin{matrix} Side & Angle \\ \begin{matrix} b = & 5 \\ c = & 13 \\ a = \end{matrix} & \begin{matrix} A = \\ B = \\ C = \end{matrix} \end{matrix}$

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{{(13)}^{2} - {(5)}^{2}}

$a = \sqrt{{(13)}^{2} - {(5)}^{2}}$

Step 2.4

Raise

13

$13$ to the power of

2

$2$ .

a = \sqrt{169 - {(5)}^{2}}

$a = \sqrt{169 - {(5)}^{2}}$

Step 2.5

Raise

5

$5$ to the power of

2

$2$ .

a = \sqrt{169 - 1 \cdot 25}

$a = \sqrt{169 - 1 \cdot 25}$

Step 2.6

Multiply

- 1

$- 1$ by

25

$25$ .

a = \sqrt{169 - 25}

$a = \sqrt{169 - 25}$

Step 2.7

Subtract

25

$25$ from

169

$169$ .

a = \sqrt{144}

$a = \sqrt{144}$

Step 2.8

Rewrite

144

$144$ as

12^{2}

$12^{2}$ .

a = \sqrt{12^{2}}

$a = \sqrt{12^{2}}$

Step 2.9

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

a = 12

$a = 12$

a = 12

$a = 12$

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

13

$13$ of the triangle.

B = \arcsin (\frac{5}{13})

$B = \arcsin (\frac{5}{13})$

Step 3.3

Evaluate

\arcsin (\frac{5}{13})

$\arcsin (\frac{5}{13})$ .

B = 22.61986494

$B = 22.61986494$

B = 22.61986494

$B = 22.61986494$

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 + 22.61986494 = 180

$A + 90 + 22.61986494 = 180$

Step 4.2

Solve the equation for

A

$A$ .

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Step 4.2.1

Add

90

$90$ and

22.61986494

$22.61986494$ .

A + 112.61986494 = 180

$A + 112.61986494 = 180$

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

112.61986494

$112.61986494$ from both sides of the equation.

A = 180 - 112.61986494

$A = 180 - 112.61986494$

Step 4.2.2.2

Subtract

112.61986494

$112.61986494$ from

180

$180$ .

A = 67.38013505

$A = 67.38013505$

A = 67.38013505

$A = 67.38013505$

A = 67.38013505

$A = 67.38013505$

A = 67.38013505

$A = 67.38013505$

Step 5

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

A = 67.38013505

$A = 67.38013505$

B = 22.61986494

$B = 22.61986494$

C = 90

$C = 90$

a = 12

$a = 12$

b = 5

$b = 5$

c = 13

$c = 13$



\begin{matrix} Side & Angle \\ \begin{matrix} b = & 5 \\ c = & 13 \\ a = & ? \end{matrix} & \begin{matrix} A = & ? \\ B = & ? \\ C = & ? \end{matrix} \end{matrix}

$\begin{matrix} Side & Angle \\ \begin{matrix} b = & 5 \\ c = & 13 \\ a = & ? \end{matrix} & \begin{matrix} A = & ? \\ B = & ? \\ C = & ? \end{matrix} \end{matrix}$

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