SOLUTION: The function f is such that f(x) = a - bcosx for 0° ≤ x ≤ 360°, where a and b are positive constants. The maximum value of f(x) is 10 and the minimum value is -2. i) Find th

Question 1202354: The function f is such that f(x) = a - bcosx for 0° ≤ x ≤ 360°, where a and b are positive constants. The maximum value of f(x) is 10 and the minimum value is -2.
i) Find the values of a and b,
ii) Solve the equation f(x)=0,
iii)Sketch the graph of y=f(x)

Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Part (i)

The max and min are 10 and -2 respectively.
The midpoint of the max and min is (10+(-2))/2 = (10-2)/2 = 8/2 = 4, which is the value of 'a'.
This is the midline.

The function updates to f(x) = 4 - b*cos(x)

Cosine maxes out when x = 0 degrees.
Cos(x) = cos(0) = 1
When cosine is maxed out, 4 - b*cos(x) will reach its minimum. In this case, the min is -2

4 - b*cos(x) = -2
4 - b*cos(0) = -2
-b*1 = -2-4
-b = -6
b = 6

Therefore, the function is f(x) = 4 - 6*cos(x)

Answers: a = 4, b = 6

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Part (ii)

f(x) = 4 - 6*cos(x)
0 = 4 - 6*cos(x)
6cos(x) = 4
cos(x) = 4/6
x = arccos(4/6) or x = -arccos(4/6)
x = 48.189685 or x = -48.189685 approximately

The angle -48.189685 is not in the interval 0° ≤ x ≤ 360°, but adding 360 to it will find a coterminal angle.
-48.189685 + 360 = 311.810315

Answers: x = 48.189685, x = 311.810315 (both are approximate)

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Part (iii)

I recommend graphing apps such as Desmos and GeoGebra.

Here's the link to the interactive Desmos graph.
https://www.desmos.com/calculator/roduixexnc

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