Question

maximum value of 8cosX-15 sinX-2​

Answer 1

Step-by-step explanation:

The summand 15 can be added later.

Let’s ask, more generally, what are the maximum and minimum values of

f(x)=acosx+bsinx

assuming a and b are not both zero. We can see that, setting c=a2+b2−−−−−−√ , we have

(ac)2+(bc)2=1

and therefore there exists a unique angle φ , with 0≤φ<2π such that

cosφ=ac,sinφ=bc

because (a/c,b/c) is a point on the unit circle. Thus we can write

f(x)=c(accosx+bcsinx)=c(cosxcosφ+sinxsinφ)=ccos(x−φ)

Thus the maximum and minimum values are c and −c .

In your case a=8 and b=15 , so

c=64+225−−−−−−−√=17

Thus the maximum and minimum values of

8cosA+15sinA+15

are, respectively, 17+15=32 and −17+15=−2 .