Grade 11 Trigonometry question help

A smokestack, AB, is 205 m high. From two points C and D on the same side of the smokestacks base B, the angles of elevation to the top of the smokestack are 40° and 36° respectively . Find the distance between C and D . It came with a diagram but I can't really show it so I'm sorry but any help would be much appreciated.

First you need to find the Sines and Cosines of Angles C (40 degrees) and D (36 degrees). You will need a sin/cosine chart, or a scientific calculator to do this:
Sin(40) = 0.64279;
Cos(40) = 0.76604;

First, find the sides for the 40 degree angle:

Step 1; find the hypotenuse BC for the 40 degree angle: (where A = bottom of stack; B = top of stack)
Sin(C) = opp/hyp; therefore hyp = opp / 0.64279 = 318.92

Step 2; find the adjacent side AC for the 40 degree angle:
Cos(C) = adj/hyp; therefore ajd = hyp * 0.76604 = 244.31

Now, repeat the sequence to find the sides for the 36 degree angle:
Sin(36) = 0.58779;
Cos(36) = 0.80902;

Step 3; find the hypotenuse BD for the 36 degree angle:

Sin(C) = opp/hyp; therefore hyp = opp / 0.58779 = 348.77

Step 4; find the adjacent side AD for the 36 degree angle:
Cos(C) = adj/hyp; therefore ajd = hyp * 0.80902 = 282.16

Side AC = 244.31 and Side AD = 282.16; the distance between C and D, and the answer, is 37.85

Hope this helps...

Let k represent the distance from B to C
Let m represent the distance from C to D

Note: D is further away from B than C is from B

There are two right triangles. The facts of the triangles are

cos(40º) = k/205 so k = 205cos(40 º)
cos(36 º) = (k + m)/205 or k + m = 205cos(36 º)

So m = 205cos(36 º) ‒ 205cos(40 º)

Can you take it from here?

tan36 = 205/L
L = 205/tan36 = 205/0.7265 = 282.2m

tan40 = 205/(L - CD)
L - CD = 205/0.8391 = 244.3m

CD = 282.2 - 244.3 = 37.9m

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