Mathfolis

Sine and Cosine Function Graphs

Unit 3 · Trigonometric and Polar Functions

What AP Precalc asks here

The graphs of y=sin⁡xy = \sin x and y=cos⁡xy = \cos x both have period 2π2\pi and range [−1,1][-1, 1]. Sine starts at (0,0)(0, 0) and is odd, so its graph is symmetric about the origin. Cosine starts at (0,1)(0, 1) and is even, so its graph is symmetric about the y-axis. The two graphs are horizontal shifts of each other by π/2\pi/2.

Properties

Period
2π for both sin and cos2\pi \text{ for both sin and cos}
Range
[−1,1] for both[-1, 1] \text{ for both}
Symmetry
sin⁡(−x)=−sin⁡x (odd);    cos⁡(−x)=cos⁡x (even)\sin(-x) = -\sin x \text{ (odd)};\;\; \cos(-x) = \cos x \text{ (even)}
AP Tip: cos⁡x=sin⁡(x+π/2)\cos x = \sin(x + \pi/2) — cosine is sine shifted left by π/2\pi/2.
Type 1

Key graph features

Use the period 2π and range [−1, 1] to answer questions about zeros, maxima, and minima.

Example 1
What is the period of y=sin⁡xy = \sin x? (A) π/2\pi/2 (B) π\pi (C) 2π2\pi (D) 4π4\pi

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Type 2

Domain, range, and symmetry

Domain is all reals; range is [−1, 1]. sin is odd; cos is even.

Example 2
Which is correct about the symmetry of sine and cosine? (A) Both are odd functions (B) Both are even functions (C) sin is odd, cos is even (D) sin is even, cos is odd

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Type 3

Match a graph to an equation

Distinguish sin from cos by checking the y-intercept (0 for sin, 1 for cos).

Example 3
A graph passes through (0,1)(0, 1), has its first zero to the right of the origin at x=π/2x = \pi/2, and reaches −1-1 at x=πx = \pi. Which function is it? (A) y=sin⁡xy = \sin x (B) y=cos⁡xy = \cos x (C) y=−sin⁡xy = -\sin x (D) y=−cos⁡xy = -\cos x

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