Trigonometry Examples

$a = 20$ , $c = 21$ , $C = 90$

Step 1

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

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Step 1.1

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 1.2

Solve the equation for $b$ .

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

Step 1.3

Substitute the actual values into the equation.

$b = \sqrt{{(21)}^{2} - {(20)}^{2}}$

Step 1.4

Raise $21$ to the power of $2$ .

$b = \sqrt{441 - {(20)}^{2}}$

Step 1.5

Raise $20$ to the power of $2$ .

$b = \sqrt{441 - 1 \cdot 400}$

Step 1.6

Multiply $- 1$ by $400$ .

$b = \sqrt{441 - 400}$

Step 1.7

Subtract $400$ from $441$ .

$b = \sqrt{41}$

Step 2

Find $B$ .

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Step 2.1

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

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

Step 2.2

Substitute in the values of the opposite side to angle $B$ and hypotenuse $21$ of the triangle.

$B = \arcsin (\frac{\sqrt{41}}{21})$

Step 2.3

Evaluate $\arcsin (\frac{\sqrt{41}}{21})$ .

$B = 17.75279016$

Step 3

Find the last angle of the triangle.

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Step 3.1

The sum of all the angles in a triangle is $180$ degrees.

$A + 90 + 17.75279016 = 180$

Step 3.2

Solve the equation for $A$ .

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Step 3.2.1

Add $90$ and $17.75279016$ .

$A + 107.75279016 = 180$

Step 3.2.2

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

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Step 3.2.2.1

Subtract $107.75279016$ from both sides of the equation.

$A = 180 - 107.75279016$

Step 3.2.2.2

Subtract $107.75279016$ from $180$ .

$A = 72.24720983$

Step 4

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

$A = 72.24720983$

$B = 17.75279016$

$C = 90$

$a = 20$

$b = \sqrt{41}$

$c = 21$