Using the trigonometric ratios, the values of x and y are: x = 8√5 and y = 8√15.
How to find the values of x and y using trigonometric ratios?
Let's find the value of the hypotenuse of the triangle where x and y lie. In the other triangle, it is one of its legs.
Using cosine ratio [cos ∅ = adj/hyp], we have:
∅ = 45°
adj = ?
hyp = 16√10
Substitute:
cos 45 = adj/16√10
cos 45 * 16√10 = adj
1/√2 * 16√10 = adj
adj = 16√10 / √2 * √2 / √2
adj = 16√5
Find x using the sine ratio:
Sin 30 = opp/hyp
sin 30 = x/16√5
sin 30 * 16√5 = x
1/2 * 16√5 = x
x = 8√5
Find y using the tangent ratio:
tan 30 = opp/adj
tan 30 = x/y
tan 30 = 8√5 / y
y * tan 30 = 8√5
y = 8√5 / tan 30
y = 8√5 / 1/√3
y = 8√5 * √3
y = 8√15