SOLUTION: Hello tutor, how do I; State the maximum and minimum values of 12cos(x) - 5sin(x) and where they occur within the range -180° < x < 180°.

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Question 738350: Hello tutor, how do I;
State the maximum and minimum values of 12cos(x) - 5sin(x) and where they occur within the range -180° < x < 180°.
Answer by Ed Parker(21) About Me  (Show Source):
You can put this solution on YOUR website!
State the maximum and minimum values of 12 cosx-5sinx and where they occur within the range -180o 
12cos(x) - 5sin(x)

Construct a right triangle with coefficients 12 and 5 for legs



Multiply and divide by 13

13%2F1312cos%28x%29%22%22-%22%2213%2F135sin%28x%29

Factor out 13

13%28expr%2812%2F13%29cos%28x%29-expr%285%2F13%29sin%28x%29%29

which becomes:

13%28cos%28theta%29cos%28x%29-sin%28theta%29sin%28x%29%29

Using the identity cos%28alpha%2Bbeta%29=cos%28alpha%29cos%28beta%29-sin%28alpha%29sin%28beta%29

13cos%28theta%2Bx%29

which we write as

13cos%28x%2Btheta%29


13cos%28x%2B%2222.61986495%B0%22%29

-180 < x < 180
-180°+22.61986495° < x+22.61986495° < 180°+22.61986495° 

-157.3801351° < x+22.61986495° < 202.6198649°

The cosine reaches a maximum value of 1 in that interval when

x+22.61986495° = 0 or
             x = -22.61986495°
             
Thus at that point 13cos%28x%2B%2222.61986495%B0%22%29 = 13

So its maximum value is 13 when x = -22.61986495° = -tan-1%285%2F12%29

---

The cosine reaches a minimum value of -1 in that interval when

x+22.61986495° = 180° or
             x = 157.3801228°
             
Thus at that point 13cos%28x%2B%2222.61986495%B0%22%29 = 13(-1) = -13

So its maximum value is -13 when x = 157.3801228° = 180°-tan-1%285%2F12%29

Edwin