(iv)

For the expression to have a maximum value, the fraction must be as positive big as possible.

This happens when the denominator 14 cos θ - 5 sin θ is as small positive as possible (like 0.000000000000000000...1).

The possible values of 14 cos θ - 5 sin θ lie between - sqrt 221 and sqrt 221.

The sad part about the denominator being as close to the positive side of zero as possible is that there is no defined maximum value (as the fraction 1/x always gets larger as x gets even closer to positive 0, with no maximum limit).

As such, we cannot determine a maximum value for this function (as does minimum value, in fact, for a very similar reason).

The ans key says min value as 14.9. But I get -14.9 for part (ii)

(i)

Using the cosine form of the R-formula (sine is also ok) and using degree form for the angles,

f(θ)
= 14 cos θ - 5 sin θ
= R times cos (θ + alpha)
= sqrt (14² + 5²) times cos (θ + taninv 5/14)
= sqrt 221 times cos (θ + 19.65382406°)

When f(θ) = 8,
sqrt 221 times cos (θ + 19.6538...°) = 8
cos (θ + 19.6538...°) = 8 / sqrt 221
cos (θ + 19.6538...°) = 0.5381382352

Basic angle, gamma (do not use alpha here)
= cosinv (0.5381382352)
= 57.44300984°

Clearly θ + 19.6538...° must lie in the first quadrant or the fourth quadrant, so

θ + 19.6538...° = gamma or 360° - gamma
θ + 19.6538...° = 57.443...° or 360° - 57.443...°
θ = 37.78918578° or 282.9031661°
θ ~ 37.8° or 282.9°

(ii)

f(θ) = sqrt 221 times cos (θ + 19.65382406°)

Minimum value of f(θ)
= sqrt 221 times (-1)
= - sqrt 221

(iii)

This minimum value occurs when

cos (θ + 19.65382406°) = -1
θ + 19.65382406° = 180°
θ = 160..34617594°
θ ~ 160.3°

I presume the author of the answer key missed out the negative sign while typing out the solutions (sometimes when you type answers in a rush, you miss out a symbol, or the system does not detect a symbol sometimes).

Thank u Mr. Eric