Answer
$b =\sqrt{21}\approx 4.58$
$A \approx 23.6°$
$B \approx 66.4°$
Work Step by Step
Since we already have the measures of two of the sides of the triangle, we can use the Pythagorean theorem to figure out $b$, one of the sides of the right triangle:
$a^2 + b^2 = c^2$
Let's plug in the values we are given:
$2^2 + b^2 = 5^2$
Evaluate the exponents:
$4 + b^2 = 25$
Subtract $4$ from each side of the equation:
$b^2 = 21$
Take the square root of both sides of the equation:
$b = \sqrt {21}\approx 4.58$
To find the value of $A$, we can use one of the trig identities. Let us use the sine identity:
$\sin A = \frac{\text{opposite}}{\text{hypotenuse}}$ = $\frac{a}{c}$
We know that $a$ is $2$ and $c$ is $5$. Let's plug in these values into our formula so that we can find $A$:
$\sin A$ = $\frac{2}{5}$
Take the inverse sine of both sides of the equation:
$A = \sin^{-1} \frac{2}{5}$
Evaluate to solve for $A$, remembering to round off to one decimal place:
$A \approx 23.6°$
To find $B$, we can use the cosine identity:
$\cos B = \frac{\text{adjacent}}{\text{hypotenuse}}$ = $\frac{a}{c}$
We plug in the values we know into this formula:
$\cos B$ = $\frac{2}{5}$
Take the inverse cosine of both sides of the equation:
$B = \cos^{-1} \frac{2}{5}$
Evaluate to solve for $B$, rounding the answer off to one decimal place:
$B \approx 66.4°$
