which of these is an equivalent representation of $f(x) = \sin(x)$?

$\sin(x + \pi)$
$\sin(x + 2\pi)$
$\cos(x + \pi)$
$\cos(x + 2\pi)$

Question

which of these is an equivalent representation of $f(x) = \sin(x)$?

$\sin(x + \pi)$
$\sin(x + 2\pi)$
$\cos(x + \pi)$
$\cos(x + 2\pi)$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Periodic Functions" ], "new_concepts": [ "Trigonometric Identities", "Phase Shifts" ], "current_concepts": [ "Periodic Functions", "Trigonometric Identities", "Phase Shifts" ] } ## Identify the target function We need to find an expression equivalent to: $$ f(x) = \sin(x) $$ ## Analyze the periodic properties Using the **Periodic Functions** knowledge point $$ \sin(x + 2\pi) = \sin(x) $$ ## Evaluate the remaining options Using the **Periodic Functions** knowledge point $$ \begin{aligned} &\sin(x + \pi) = -\sin(x) \ &\cos(x + \pi) = -\cos(x) \ &\cos(x + 2\pi) = \cos(x) \end{aligned} $$ ## Match with the correct option The only expression that simplifies exactly to $\sin(x)$ is $\sin(x + 2\pi)$.

$\sin(x + \pi)$

$\sin(x + 2\pi)$

$\cos(x + \pi)$

$\cos(x + 2\pi)$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Periodic Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the target function

Analyze the periodic properties

Evaluate the remaining options

Match with the correct option

Answer: