Question

Arrange the functions in decreasing order of their periods:

a) y=−3cos(x+2π)
b) y= 2/3 cot( x/4 )+6
c) y=− 1/2 tan( 5x/6+π)
d) y=5

Answer 1

The functions arranged in decreasing order of their periods are y=5 (infinite period), y=2/3 cot(x/4)+6 (8π), y=-1/2 tan(5x/6+π) (6π/5), and y=-3cos(x+2π) (2π).

Step-by-step explanation:

The student has asked to arrange the functions in decreasing order of their periods. To determine the period of trigonometric functions, we need to look at the argument of the sine, cosine, or tangent functions. Remember that the standard period of cosine and sine functions is 2π, and for cotangent and tangent functions, the period is π. The period (T) of the function can be found by taking 2π divided by the absolute value of the coefficient of x in the argument (for cosine and sine) or π divided by the absolute value of the coefficient of x (for cotangent and tangent).

• y=-3cos(x+2π) has the standard period of 2π because the coefficient of x is 1.
• y=2/3 cot(x/4)+6 has a period of 2π divided by 1/4, which is 8π.
• y=-1/2 tan(5x/6+π) has a period of π divided by 5/6, which is 6π/5.
• y=5 is a constant function and thus does not have a periodic behavior; its period is considered infinite.

Arranging these periods from largest to smallest gives: y=5, y=2/3 cot(x/4)+6, y=-1/2 tan(5x/6+π), and y=-3cos(x+2π).