Question

So i get the first few parts but i don’t know how to answer the last part, could anyone help please?

i) Sketch, on a single diagram showing values of x from-180° to +180°, the graphs of
y = tan x and y = 4 cos x.

The equation : tan x = 4 cos x
has two roots in the interval -180° ≤x≤ 180°. These are denoted by a and ß, where a <ß.

ii) Show a and ß on your sketch, and express ß in terms of a

iii) Show that the equation tan x = 4 cos x may be written as 4 sin^2x+ sin x−4=0.
Hence find the value of ß-a, correct to the nearest degree.

Answer 1

i) The graph of y = tan x will be a periodic function with period of 180 degrees, with asymptotes at x = (n+1/2)π, where n is an integer. The graph of y = 4 cos x will be a cosine function with amplitude 4 and period 360 degrees.

ii) The roots of the equation tan x = 4 cos x, denoted by a and ß, can be found by solving the equation tan x = 4 cos x for x. The two solutions will be a and ß, where a <ß.

iii) The equation tan x = 4 cos x can be written as 4 sin^2x+ sin x−4=0 by using trigonometric identities. The value of ß-a can be found by using the quadratic formula on this equation. However, it's not possible to give you an exact value without the specific value of a.