DWG NO. 4.4 — Lesson 4 of 6

Graphing Sine and Cosine Functions

Unit 4: Trigonometric Functions and the Unit Circle · ~20 min

Objective Graph y = sin x and y = cos x over one full period, identifying key points, domain, range, and period.

As θ sweeps counterclockwise around the unit circle, sin θ (the y-coordinate of the point) rises and falls smoothly, and the whole pattern repeats every 2π radians — one full trip around the circle. Plotting sin θ against θ turns that circular motion into the familiar wave shape. Cosine does exactly the same thing, just tracking the x-coordinate instead.

Key features of y = sin x and y = cos x

Feature	y = sin x	y = cos x
Domain	(−∞, ∞)	(−∞, ∞)
Range	[−1, 1]	[−1, 1]
Period	2π	2π
x-intercepts	x = nπ	x = \(\frac{\pi}{2}\) + nπ
y-intercept	(0, 0)	(0, 1)

Both graphs are bounded between −1 and 1 — a direct consequence of both being coordinates on a circle of radius 1 — and both are periodic, meaning the entire shape repeats forever in both directions.

^π/2 π 3π/2 2π y = sin x y = cos x

y = sin x (solid) and y = cos x (dashed) over one full period, 0 to 2π — cos x is sin x shifted left by \(\frac{\pi}{2}\)

Worked Example 1 · Key points of y = sin x

ProblemList the key points of y = sin x on [0, 2π].

1Using the unit circle: sin 0 = 0, sin \(\frac{\pi}{2}\) = 1, sin π = 0, sin \(\frac{3\pi}{2}\) = −1, sin 2π = 0.

2These give a zero, a maximum, a zero, a minimum, and a zero — the shape of one full wave.

(0,0), (\(\frac{\pi}{2}\),1), (π,0), (\(\frac{3\pi}{2}\),−1), (2π,0)

Worked Example 2 · Reading y = cos x from a value

ProblemWithout a calculator, find cos π and describe what point on the graph that gives.

1At θ = π, the unit-circle point is (−1, 0), so cos π = −1 — the x-coordinate.

2On the graph of y = cos x, this is the point (π, −1) — the graph's minimum.

cos π = −1, giving the minimum point (π, −1)

Guided practice

1.State the domain and range of y = cos x.
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Domain: (−∞, ∞). Range: [−1, 1].
2.Find all x-intercepts of y = sin x on the interval [0, 2π].
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sin x = 0 at x = 0, π, and 2π on this interval.
3.What is the maximum value of y = sin x, and at what x-value (in [0, 2π]) does it occur?
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Maximum value 1, occurring at x = \(\frac{\pi}{2}\).
4.Explain why sin x and cos x have the same range, using the unit-circle definition.
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Both are coordinates of a point on the unit circle (radius 1), so both are bounded between −1 and 1 for the same geometric reason.
5.Describe how the graph of y = cos x relates to the graph of y = sin x.
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y = cos x is y = sin x shifted left by \(\frac{\pi}{2}\) — same wave shape, same period and range, just starting at its maximum instead of at zero.

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