Trigonometry Examples

$c = 20$ , $a = 3.38$ , $c = 6.55$

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 $b$ .

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

Step 2.3

Substitute the actual values into the equation.

$b = \sqrt{{(6.55)}^{2} - {(3.38)}^{2}}$

Step 2.4

Raise $6.55$ to the power of $2$ .

$b = \sqrt{42.9025 - {(3.38)}^{2}}$

Step 2.5

Raise $3.38$ to the power of $2$ .

$b = \sqrt{42.9025 - 1 \cdot 11.4244}$

Step 2.6

Multiply $- 1$ by $11.4244$ .

$b = \sqrt{42.9025 - 11.4244}$

Step 2.7

Subtract $11.4244$ from $42.9025$ .

$b = \sqrt{31.4781}$

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 $6.55$ of the triangle.

$B = \arcsin (\frac{\sqrt{31.4781}}{6.55})$

Step 3.3

Evaluate the root.

$B = \arcsin (\frac{5.61053473}{6.55})$

Step 3.4

Divide $5.61053473$ by $6.55$ .

$B = \arcsin (0.85657018)$

Step 3.5

Evaluate $\arcsin (0.85657018)$ .

$B = 58.93363775$

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.

$A + 90 + 58.93363775 = 180$

Step 4.2

Solve the equation for $A$ .

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Step 4.2.1

Add $90$ and $58.93363775$ .

$A + 148.93363775 = 180$

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 $148.93363775$ from both sides of the equation.

$A = 180 - 148.93363775$

Step 4.2.2.2

Subtract $148.93363775$ from $180$ .

$A = 31.06636224$

Step 5

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

$A = 31.06636224$

$B = 58.93363775$

$C = 90$

$a = 3.38$

$b = \sqrt{31.4781}$

$c = 6.55$