Trigonometry

Question

It is given that f(x)= a cos(bx)+c , where a,b,c are integers.The maximum value of f is 11,the minimum value of f is 3 and the period of f is 72degree. Find the value of a,b,c.

Answer 1

Since cosine ranges from -1 to 1, and f varies from 3 to 11, that means that coefficient a must be multiplying the double amplitude (2 ) of a regular cosine graph by a factor of a=4 to achieve this vertical difference of 8 (11-3).

 

YOU can figure that 72 degrees is one fifth of the regular cosine's period of 360 degrees.  Thus b = ____.

 

Answer 2

y = Acos(Bx)+C

 

Since the y-values vary from 3 to 11, the amplitude, A, is half the difference between the maximum and minimum values.

 

So, A = 4

 

72° = 72°(π/180) = 2π/5   Period = 2π/B = 2π/5  So, B = 5.

 

Since the amplitude is 4, if there were no vertical shift, the y values would vary from -4 to 4.  However, the y values vary from 3 to 11, so the graph was shifted upward by 7 units.  Thus, C = 7.

 

Equation:  y = 4cos(5x) + 7