Textbook Question
In Exercises 39–42, express the given quantity in terms of sin x and cos x.
sin (2π − x)
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0. Functions
Introduction to Trigonometric Functions
Multiple Choice
Which of the following is the period of the function with respect to ?
A
B
There is no period
C
D
Verified step by step guidance1
Step 1: Recall the definition of the period of a function. The period of a function is the smallest positive value T such that the function repeats itself, i.e., f(x + T) = f(x).
Step 2: Consider the given function z = sin(x - y). The sine function has a fundamental period of 2π, meaning it repeats every 2π units.
Step 3: Observe that the variable y does not affect the periodicity with respect to x because it is treated as a constant when analyzing the function's behavior in terms of x.
Step 4: Substitute x with (x + 2π) in the function: z = sin((x + 2π) - y). Simplify the argument of the sine function: (x + 2π - y) = (x - y) + 2π.
Step 5: Use the property of the sine function, sin(a + 2π) = sin(a), to confirm that z = sin(x - y) repeats itself every 2π with respect to x. Therefore, the period of the function with respect to x is 2π.
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