Answer
$c \approx 5.83$
$A \approx 59.0°$
$B \approx 31.0°$
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 $c$, the length of the hypotenuse:
$a^2 + b^2 = c^2$
Let's plug in the values we are given:
$5^2 + 3^2 = c^2$
Evaluate the exponents:
$25 + 9 = c^2$
Add on the left side of the equation:
$34 = c^2$
Take the square root of both sides of the equation:
$c = \sqrt {34}$
Evaluate the square root, rounding off to two decimal places:
$c \approx 5.83$
To find the value of $A$, we can use one of the trig identities. Let's use the tangent identity:
$\tan A = \frac{opposite}{adjacent}$ = $\frac{a}{b}$
We know that $a$ is $5$ and $b$ is $3$. Let's plug in these values into our formula so that we can find $A$:
$\tan A = \frac{5}{3}$
Take the inverse tangent of both sides of the equation:
$A = \tan^{-1} \left(\frac{5}{3}\right)$
Evaluate to solve for $A$, remembering to round off to one decimal place:
$A \approx 59.0°$
To find $B$, we can use the tangent identity again:
$\tan B = \frac{opposite}{adjacent}$ = $\frac{b}{a}$
We plug in the values we know into this formula:
$\tan B = \frac{3}{5}$
Take the inverse tangent of both sides of the equation:
$B = \tan^{-1} \left(\frac{5}{3}\right)$
Evaluate to solve for $B$, rounding the answer off to one decimal place:
$B \approx 31.0°$
