The angles of elevation from the two points to the object

Question

Questions -

Q1. Prove the following identities

a. sin(x + y) + sin(x - y) = 2 sin x cos y

b. sec(x - y) = cos(x + y)/(cos²x - sin²y)

c. tan²x - sin²x = (tan x sin x)²

Q2. Solve the following equations for x ∈ [0, 2π] and verify the result using MATLAB.

a. tan x = cot x

b. 2 sin x + cot x - csc x = 0

c. 18 sec²x - 17 tan x sec x - 12 = 0

Q3. Solve the following problems

a. A person standing in front of a building with a height of 50 m finds that the angle of elevation to the top of the building is 38^o. He then walks towards the building x m and finds the new angle of elevation to be 46^o. Calculate x.

b. A flying object is observed from two points on ground level that are on opposite sides of the object are 10 m apart. The angles of elevation from the two points to the object are 30^o and 40^o. Calculate the height of the object above the ground.

References -

1. McKeague, C. P., & Turner, M. D. (2013). Trigonometry. Australia: Brooks/Cole, Cengage Learning.

2. Lial, M. L. (2013). Trigonometry. Boston: Pearson.