Trigonometry Examples

$a = 4$ , $c = 2$ , $B = 90$

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.

$b = \sqrt{{(2)}^{2} + {(4)}^{2}}$

Step 1.4

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

$b = \sqrt{4 + {(4)}^{2}}$

Step 1.5

Raise $4$ to the power of $2$ .

$b = \sqrt{4 + 16}$

Step 1.6

Add $4$ and $16$ .

$b = \sqrt{20}$

Step 1.7

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 1.7.1

Factor $4$ out of $20$ .

$b = \sqrt{4 (5)}$

Step 1.7.2

Rewrite $4$ as $2^{2}$ .

$b = \sqrt{2^{2} \cdot 5}$

Step 1.8

Pull terms out from under the radical.

$b = 2 \sqrt{5}$

Step 2

Find $C$ .

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Step 2.1

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

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

Step 2.2

Substitute in the values of the opposite side to angle $C$ and hypotenuse $2 \sqrt{5}$ of the triangle.

$C = \arcsin (\frac{2}{2 \sqrt{5}})$

Step 2.3

Cancel the common factor of $2$ .

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Step 2.3.1

Cancel the common factor.

$C = \arcsin (\frac{2}{2 \sqrt{5}})$

Step 2.3.2

Rewrite the expression.

$C = \arcsin (\frac{1}{\sqrt{5}})$

Step 2.4

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$C = \arcsin (\frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Step 2.5

Combine and simplify the denominator.

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Step 2.5.1

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$C = \arcsin (\frac{\sqrt{5}}{\sqrt{5} \sqrt{5}})$

Step 2.5.2

Raise $\sqrt{5}$ to the power of $1$ .

$C = \arcsin (\frac{\sqrt{5}}{\sqrt{5} \sqrt{5}})$

Step 2.5.3

Raise $\sqrt{5}$ to the power of $1$ .

$C = \arcsin (\frac{\sqrt{5}}{\sqrt{5} \sqrt{5}})$

Step 2.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$C = \arcsin (\frac{\sqrt{5}}{{\sqrt{5}}^{1 + 1}})$

Step 2.5.5

Add $1$ and $1$ .

$C = \arcsin (\frac{\sqrt{5}}{{\sqrt{5}}^{2}})$

Step 2.5.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$C = \arcsin (\frac{\sqrt{5}}{{(5^{\frac{1}{2}})}^{2}})$

Step 2.5.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$C = \arcsin (\frac{\sqrt{5}}{5^{\frac{1}{2} \cdot 2}})$

Step 2.5.6.3

Combine $\frac{1}{2}$ and $2$ .

$C = \arcsin (\frac{\sqrt{5}}{5^{\frac{2}{2}}})$

Step 2.5.6.4

Cancel the common factor of $2$ .

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Step 2.5.6.4.1

Cancel the common factor.

$C = \arcsin (\frac{\sqrt{5}}{5^{\frac{2}{2}}})$

Step 2.5.6.4.2

Rewrite the expression.

$C = \arcsin (\frac{\sqrt{5}}{5})$

Step 2.5.6.5

Evaluate the exponent.

$C = \arcsin (\frac{\sqrt{5}}{5})$

Step 2.6

Evaluate $\arcsin (\frac{\sqrt{5}}{5})$ .

$C = 26.56505117$

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.

$A + 26.56505117 + 90 = 180$

Step 3.2

Solve the equation for $A$ .

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Step 3.2.1

Add $26.56505117$ and $90$ .

$A + 116.56505117 = 180$

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 $116.56505117$ from both sides of the equation.

$A = 180 - 116.56505117$

Step 3.2.2.2

Subtract $116.56505117$ from $180$ .

$A = 63.43494882$

Step 4

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

$A = 63.43494882$

$B = 90$

$C = 26.56505117$

$a = 4$

$b = 2 \sqrt{5}$

$c = 2$