Question

In right triangle abc below where ac=2 and angle BAC is theta find y in terms of theta

please ignore my work!​

Answer 1

2tanθ -2 = y

Explanation:

First, to make things a little simpler, we can say that the point between C and B is D, so BD = y

We are given that ∠ADC = 45° , and ∠ACD = 90°. Because the angles of a triangle add up to 180 degrees, we can say

∠CAD + ∠ACD + ∠ADC = 180°

∠CAD + 90 + 45 = 180

∠CAD = 45°

In a 45/45/90 degree triangle, we can say that the two non-hypotenuse sides are the same. The hypotenuse is opposite of the right angle, so in this triangle, it would be AD. Therefore, we can say that AC = CD = 2

Because BC = CD + DB , we can say that

BC = CD + DB

BC = 2 + y

y = BC -2

Therefore, if we know BC, we can solve for y.

We want to solve for BC given

- the angle opposite of it (θ)

- the side adjacent to the angle (AC).

One equation that fits this is

tanθ = opposite/adjacent

tanθ = BC/AC

tanθ = (2+y)/2

multiply both sides by 2 to remove a denominator

2 tanθ=2+y

subtract 2 from both sides to isolate th y

2tanθ -2 = y