SOLUTION: "show that the maximum and minimum values of the function {{{f(theta) = 5cos(theta) + 12sin(theta)}}} are 13 and -13 respectively" plzz can anyone help me wd this...

Algebra ->  Trigonometry-basics -> SOLUTION: "show that the maximum and minimum values of the function {{{f(theta) = 5cos(theta) + 12sin(theta)}}} are 13 and -13 respectively" plzz can anyone help me wd this...      Log On


   



Question 895132: "show that the maximum and minimum values of the function f%28theta%29+=+5cos%28theta%29+%2B+12sin%28theta%29 are 13 and -13 respectively" plzz can anyone help me wd this...
Answer by Edwin McCravy(20065) About Me  (Show Source):
You can put this solution on YOUR website!

f%28theta%29+=+5cos%28theta%29+%2B+12sin%28theta%29

%22f%27%22%28theta%29+=+-5sin%28theta%29%2B12cos%28theta%29

Set that = 0 to find first coordinates of possible 
maximum and minimum points:

-5sin%28theta%29%2B12cos%28theta%29=0

-5sin%28theta%29=-12cos%28theta%29

Divide both sides by -5cos%28theta%29



sin%28theta%29%2Fcos%28theta%29=+12%2F5

tan%28theta%29+=+12%2F5

Since tangent = y%2Fx = %28-y%29%2F%28-x%29, let y=±12, x=±5 

r=sqrt%28x%5E2%2By%5E2%29

r=sqrt%28%28%22%22+%2B-+5%29%5E2%2B+%28%22%22+%2B-+12%29%5E2%29

r=sqrt%2825%2B144%29

r=sqrt%28169%29

r=13

f%28theta%29+=+5cos%28theta%29+%2B+12sin%28theta%29

Now we substitute both possible values to find y-coordinates
of maximum and minimum points:

cos%28theta%29+=+x%2Fr+=+%28%22%22+%2B-+5%29%2F%28%22%22+%2B-+13%29 and 

sin%28theta%29+=+y%2Fr+=+%28%22%22+%2B-+12%29%2F%28%22%22+%2B-+13%29

f%28theta%29+=+5cos%28theta%29+%2B+12sin%28theta%29 becomes:

f%28theta%29+=+5%28%22%22+%2B-+5%2F13%29+%2B+12%28%22%22+%2B-+12%2F13%29

f%28theta%29+=+%28%22%22+%2B-+25%2F13%29+%2B+%28%22%22+%2B-+144%2F13%29

Now we choose both possible combinations of signs:

(1)  f%28theta%29+=+25%2F13+%2B+144%2F13+=+169%2F13+=+13

(2)  f%28theta%29+=+-25%2F13+-+144%2F13+=+-169%2F13+=+-13


(1) gives the maximum value and (2) gives the minimum value
Edwin