HOW TO FIND MAXIMUM AND MINIMUM VALUES OF SINE AND COSINE FUNCTION

Identify the maximum and minimum values and zeroes of the function in the interval [-2π, 2π]. Use your understanding of transformations.

Problem 1 :

y = 2sinx

Solution :

For a sine function the minimum value is -1 and maximum value is 1.

-1 ≤ sinx ≤ 1

Multiply it by 2

-2 ≤ 2sinx ≤ 2

Maximum at y = -2 and minimum at y = 2

To find for what value of x, we will have the maximum value and minimum value, we should equate

So maximum is 2 at -3π/2 and π/2 and minimum is -2 at π/2 and 3π/2.

To get zeroes :

2sinx = 0

sinx = 0

x = sin^-1(0)

x = 0, ±π, ±2π

So zeroes are 0, ±π, ±2π.

Problem 2 :

y = 3cos(x/2)

Solution :

For a cosine function the minimum value is -1 and maximum value is 1.

-1 ≤ cos(x/2) ≤ 1

Multiply it by 3

-3 ≤ 3cos(x/2) ≤ 3

Maximum at y = -3 and minimum at y = 3

To find for what value of x, we will have the maximum value and minimum value, we should equate

So maximum is 3 at 0 and 2π and minimum is -3 at π.

To get zeroes :

3cos(x/2) = 0

cos(x/2) = 0

x/2 = cos^-1(0)

x = ±π/2, ±3π/2

So zeroes are ±π/2, ±3π/2.

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