Which of the following are equivalent to the function [tex]$y=3 \cos x+2$[/tex]? Check all that apply.
A. [tex]$y=-3 \cos x-2$[/tex]
B. [tex]$y=3 \sin \left(x+\frac{\pi}{2}\right)+2$[/tex]
C. [tex]$y=3 \cos (-x)+2$[/tex]
D. [tex]$y=3 \sin \left(x-\frac{\pi}{2}\right)+2$[/tex]
Answer (1)
Option A is not equivalent because it has opposite signs.
Option B is equivalent because sin ( x + 2 π ) = cos x .
Option C is equivalent because cos ( − x ) = cos x .
Option D is not equivalent because sin ( x − 2 π ) = − cos x .
Therefore, the equivalent options are B , C .
Explanation
Analyzing the Problem We are given the function y = 3 cos x + 2 and we need to determine which of the options are equivalent to it. Let's analyze each option.
Analyzing Option A Option A: y = − 3 cos x − 2 . This is not equivalent to y = 3 cos x + 2 because the signs of both terms are different.
Analyzing Option B Option B: y = 3 sin ( x + 2 π ) + 2 . Using the trigonometric identity sin ( x + 2 π ) = cos x , we can rewrite this as y = 3 cos x + 2 , which is equivalent to the given function.
Analyzing Option C Option C: y = 3 cos ( − x ) + 2 . Using the trigonometric identity cos ( − x ) = cos x , we can rewrite this as y = 3 cos x + 2 , which is equivalent to the given function.
Analyzing Option D Option D: y = 3 sin ( x − 2 π ) + 2 . Using the trigonometric identity sin ( x − 2 π ) = − cos x , we can rewrite this as y = − 3 cos x + 2 , which is not equivalent to the given function.
Conclusion Therefore, the options that are equivalent to the function y = 3 cos x + 2 are B and C.
Examples
Understanding trigonometric functions and their transformations is crucial in many fields, such as physics and engineering. For example, when analyzing alternating current (AC) circuits, the voltage and current are often modeled as sinusoidal functions. Knowing how to manipulate these functions using trigonometric identities allows engineers to simplify circuit analysis and design more efficient systems. Similarly, in signal processing, trigonometric functions are used to represent and analyze signals, and understanding their properties is essential for filtering and manipulating these signals effectively.
