Hard qn on Trigonometric equations and identities, please help

(i) , (ii) , (iii) The maximum value of OE is approximately 1.523 meters, which is associated with an angle of approximately 23.199º.

Step-by-step explanation:

(i) From Geometry, we get that sum of internal angles of trangle AOD.

(1)

If we know that and , then the value of is:

(2)

But we also have the following identity:

(3)

If we know that and , then the value of is:

(4)

By Trigonometry, we derive the following formula:

(5)

If we know that , , and , then the value of OE is:

(6)

(ii) If we know that , then the value of :

By trial and error, we find that .

(iii) Let , the first and second derivatives of the function are, respectively:

(7)

(8)

We equalize the first derivative of the function to zero and solve for :

And we evaluate the second derivative:

Then, the critical value is associated with an absolute maximum.

The maximum value of OE is:

The maximum value of OE is approximately 1.523 meters, which is associated with an angle of approximately 23.199º.