Find maximum length of PQ as t varies

[Antroph1c]

Hello, I have a question about how to find the maximum length of PQ, the answer for question (i) is attached below, i tried finding the mark scheme for this question and I mainly didn't quite understand how to get the inequality -25<20c+15s<25

 

[blamocur]

[imath]20c + 15s = 25 \left(\frac{4}{5}c + \frac{3}{5}s\right)[/imath], but [imath]\left(\frac{4}{5}\right)^2 + \left(\frac{3}{5}\right)^2 = 1[/imath].
Let me know whether you can progress from here.

 

[Antroph1c]

Sorry but i don't quite understand what's the thought process behind that equation that you mentioned being used

 

[blamocur]

Sorry but i don't quite understand what's the thought process behind that equation that you mentioned being used

When a sum of squres of two numbers is 1 they can be represented as sine and cosine of the same angle:
[imath]20 \cos t + 15 \sin t = 25 (\sin\alpha \cos t + \cos \alpha \sin t)[/imath]