Question

1. choose the best answer. this graph represents which function? graph of a trigonometric function options: cosine, tangent, secant, cotangent

Explanation:

1. Recall the properties of trigonometric functions:

• The cosine function \( y = \cos(x) \) has a period of \( 2\pi \), starts at \( (0, 1) \) (since \( \cos(0)=1 \)), and has a smooth, wave - like curve.
• The tangent function \( y = \tan(x) \) has vertical asymptotes at \( x=\frac{\pi}{2}+k\pi,k\in\mathbb{Z} \) and is not defined at those points, so its graph has breaks, which does not match the given graph.
• The secant function \( y=\sec(x)=\frac{1}{\cos(x)} \) also has vertical asymptotes at \( x = \frac{\pi}{2}+k\pi,k\in\mathbb{Z} \) (where \( \cos(x) = 0 \)) and its graph has breaks, so it does not match.
• The cotangent function \( y=\cot(x)=\frac{\cos(x)}{\sin(x)} \) has vertical asymptotes at \( x = k\pi,k\in\mathbb{Z} \) (where \( \sin(x)=0 \)) and its graph has breaks, so it does not match.

1. Analyze the given graph:

• The given graph is a smooth, continuous wave - like curve with a period of \( 2\pi \) and passes through \( (0, 1) \), which is consistent with the graph of the cosine function \( y = \cos(x) \).

Answer:

A. cosine