Question

find the value of x & y
3
4
65°
x°
y

Explanation:

Step1: Calculate angle $x$

$\tan(x) = \frac{8}{4} = 2$
$x = \arctan(2) \approx 63.43^\circ$

Step2: Calculate side $y$ (Pythagorean theorem)

$$y = \sqrt{8^2 + 4^2} = \sqrt{64 + 16} = \sqrt{80} = 4\sqrt{5} \approx 8.94$$

Step3: Verify angle sum (optional check)

$90^\circ + 65^\circ + 63.43^\circ \approx 218.43^\circ$ (correction: use triangle angle sum for $x$: $x = 90^\circ - 65^\circ = 25^\circ$; recalculate $\tan(x)=\frac{8}{4}$ was reversed: $\tan(x)=\frac{4}{8}=0.5$)

Step1 (corrected): Find angle $x$

$x = 90^\circ - 65^\circ = 25^\circ$

Step2 (corrected): Calculate side $y$

$$y = \frac{8}{\cos(65^\circ)} \approx \frac{8}{0.4226} \approx 18.93$$
or $$y = \frac{4}{\sin(65^\circ)} \approx \frac{4}{0.9063} \approx 4.41$$ (correction: adjacent to $65^\circ$ is 4, opposite is 8)

Step1 (final): Calculate $x$ via angle sum

$x = 180^\circ - 90^\circ - 65^\circ = 25^\circ$

Step2 (final): Calculate $y$ via Pythagoras

$$y = \sqrt{8^2 + 4^2} = \sqrt{80} = 4\sqrt{5} \approx 8.94$$

Answer:

$x = 25^\circ$, $y = 4\sqrt{5}$ (or approximately $8.94$)