Questions: Graphing Sine and Cosine

Question 1

The graph of y = cos(x) can be obtained from y = sin(x) by:

• A. Shifting left by π/2
• B. Shifting right by π/2
• C. Reflecting over the x-axis
• D. Stretching vertically by a factor of 2

Answer: (A)

cos(x) = sin(x + π/2), so cosine is sine shifted LEFT by π/2. This trips up many students because the '+' inside the function means shift left (the graph reaches its peak earlier). A shift right would be sin(x − π/2), which equals −cos(x), not cos(x).

Question 2

The graph of y = sin(x) has sharp corners at its maximum and minimum values.

• T. True
• F. False

Answer: False

This is a very common drawing error. The sine function is differentiable everywhere — its derivative is cos(x), which exists at every point including peaks and troughs. At x = π/2 (a maximum), the derivative cos(π/2) = 0, so the tangent is horizontal. The graph is smooth with a flat top at each maximum and minimum, not a sharp corner.

Question 3

How does the unit circle connect to the graph of y = sin(x)? What do you read off the circle to get each value?

Model answer: For angle x, the point on the unit circle is (cos x, sin x). The y-coordinate of this point equals sin(x). As x increases from 0 to 2π, tracing around the circle, the y-coordinate traces out exactly one full period of the sine wave.

The unit circle is the source of truth for sine and cosine values. At angle x, the terminal point on the unit circle has coordinates (cos x, sin x). Reading the y-coordinate gives sin(x) and the x-coordinate gives cos(x). The five key points — (0,0), (π/2,1), (π,0), (3π/2,−1), (2π,0) — come directly from the circle's top, right, bottom, and left positions.