Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 24 \\ c = & 26 \\ a = \end{matrix} & \begin{matrix} A = \\ B = \\ C = \end{matrix} \end{matrix}$

Step 1

Assume that angle $C = 90$ .

$C = 90$

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).

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

Step 2.2

Solve the equation for $a$ .

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

Step 2.3

Substitute the actual values into the equation.

$a = \sqrt{{(26)}^{2} - {(24)}^{2}}$

Step 2.4

Raise $26$ to the power of $2$ .

$a = \sqrt{676 - {(24)}^{2}}$

Step 2.5

Raise $24$ to the power of $2$ .

$a = \sqrt{676 - 1 \cdot 576}$

Step 2.6

Multiply $- 1$ by $576$ .

$a = \sqrt{676 - 576}$

Step 2.7

Subtract $576$ from $676$ .

$a = \sqrt{100}$

Step 2.8

Rewrite $100$ as $10^{2}$ .

$a = \sqrt{10^{2}}$

Step 2.9

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

$a = 10$

Step 3

Find $B$ .

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Step 3.1

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

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

Step 3.2

Substitute in the values of the opposite side to angle $B$ and hypotenuse $26$ of the triangle.

$B = \arcsin (\frac{24}{26})$

Step 3.3

Cancel the common factor of $24$ and $26$ .

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Step 3.3.1

Factor $2$ out of $24$ .

$B = \arcsin (\frac{2 (12)}{26})$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Factor $2$ out of $26$ .

$B = \arcsin (\frac{2 \cdot 12}{2 \cdot 13})$

Step 3.3.2.2

Cancel the common factor.

$B = \arcsin (\frac{2 \cdot 12}{2 \cdot 13})$

Step 3.3.2.3

Rewrite the expression.

$B = \arcsin (\frac{12}{13})$

Step 3.4

Evaluate $\arcsin (\frac{12}{13})$ .

$B = 67.38013505$

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 $180$ degrees.

$A + 90 + 67.38013505 = 180$

Step 4.2

Solve the equation for $A$ .

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Step 4.2.1

Add $90$ and $67.38013505$ .

$A + 157.38013505 = 180$

Step 4.2.2

Move all terms not containing $A$ to the right side of the equation.

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Step 4.2.2.1

Subtract $157.38013505$ from both sides of the equation.

$A = 180 - 157.38013505$

Step 4.2.2.2

Subtract $157.38013505$ from $180$ .

$A = 22.61986494$

Step 5

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

$A = 22.61986494$

$B = 67.38013505$

$C = 90$

$a = 10$

$b = 24$

$c = 26$