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Find max and min values of a trigonometry function

Question:

Find the maximum and minimum values of the trigonometry function y = cos2x - 2 sin x + 1

Solution:

using the formula: sin2x + cos2x = 1, then cos2x = 1 - sin2x

y = cos2x - 2 sin x + 1
= 1 - sin2x - 2 sin x + 1
= -sin2x - 2 sin x + 2
= -(sin2x + 2 sin x) + 2
= -(sin2x + 2 sin x + 1 - 1) + 2
= -(sin2x + 2 sin x + 1) + 1 + 2
= -(sin x + 1)2 + 3

when sin x = -1, y = -(-1 + 1)2 + 3 = -02 + 3 = 3

when sin x = 1, y = -(1 + 1)2 + 3 = -4 + 3 = -1

Therefore, the maximum value of the function y is 3 and the minimum value of the function y is -1.

how to find the maximum and minimum values of a trigonometry function

Look the graph above, the orange curve is the graph of y = cos2x - 2 sin x + 1. The blue curve is the graph of y = sin x. When the blue curve reaches the minimum value, the orange curve reaches the maximum value 3. When the blue curve reaches the maximum value, the orange curve reaches the minimum value -1. Watch the video for more details.

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