From Unit Circle to Graphs: Basic Sine and Cosine Patterns

How sine and cosine “move” as the angle moves

On the unit circle, each angle θ corresponds to a point (x, y) on the circle of radius 1. As θ increases, that point travels counterclockwise. The key idea for graphs is this: cosine tracks the horizontal coordinate x, and sine tracks the vertical coordinate y. So as the point moves around the circle, the x-value and y-value rise and fall in a predictable, repeating way.

Think of a moving dot on the circle and imagine two “shadows” of that dot: one shadow on the x-axis (cos θ) and one shadow on the y-axis (sin θ). When the dot is near the right side of the circle, the x-shadow is large and positive; when it’s near the left side, the x-shadow is large and negative. Similarly, when the dot is near the top, the y-shadow is large and positive; near the bottom, it’s large and negative.

Amplitude and why it is 1 for basic sine and cosine

Because the unit circle has radius 1, the x- and y-coordinates of any point on it must stay between −1 and 1. That means:

• −1 ≤ cos θ ≤ 1
• −1 ≤ sin θ ≤ 1

The maximum height of the graph above 0 is 1 and the minimum is −1, so the amplitude (half the total vertical range) is 1 for both y = sin θ and y = cos θ.

Period and why it is 2π for basic sine and cosine

After one full revolution around the circle, the moving point returns to the same location. One full revolution is 2π radians, so the patterns repeat every 2π:

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• sin(θ + 2π) = sin θ
• cos(θ + 2π) = cos θ

This repeating length is the period. For the basic (unscaled) functions, the period is 2π.

From circle positions to key graph points on 0 to 2π

To graph over 0 to 2π, focus on special angles where the unit-circle coordinates are well known. Each special angle gives an exact sine and cosine value, which becomes an exact point on the graph.

Key intercepts and extreme points

These come directly from the four “axis” points on the unit circle:

• At θ = 0: (cos θ, sin θ) = (1, 0) so cos 0 = 1 and sin 0 = 0
• At θ = π/2: (0, 1) so cos(π/2) = 0 and sin(π/2) = 1
• At θ = π: (−1, 0) so cos π = −1 and sin π = 0
• At θ = 3π/2: (0, −1) so cos(3π/2) = 0 and sin(3π/2) = −1
• At θ = 2π: (1, 0) so cos(2π) = 1 and sin(2π) = 0

From these, you can read key features:

• y = sin θ crosses the θ-axis at 0, π, 2π; reaches a maximum of 1 at π/2; reaches a minimum of −1 at 3π/2.
• y = cos θ starts at 1 when θ = 0; crosses the θ-axis at π/2 and 3π/2; reaches −1 at π; returns to 1 at 2π.

A structured routine for graphing sine and cosine

Routine step 1: Make a key-angle table

Use angles that divide the circle into familiar pieces. A standard choice is:

• 0, π/6, π/4, π/3, π/2, 2π/3, 3π/4, 5π/6, π, 7π/6, 5π/4, 4π/3, 3π/2, 5π/3, 7π/4, 11π/6, 2π

Then fill in sin θ and cos θ from the unit-circle coordinates.

θ        0   π/6   π/4   π/3   π/2   2π/3  3π/4  5π/6   π   7π/6  5π/4  4π/3  3π/2  5π/3  7π/4  11π/6  2π  
cos θ    1  √3/2  √2/2   1/2    0    -1/2  -√2/2 -√3/2 -1  -√3/2 -√2/2  -1/2    0     1/2   √2/2   √3/2   1  
sin θ    0   1/2  √2/2  √3/2   1    √3/2   √2/2  1/2    0   -1/2  -√2/2 -√3/2  -1   -√3/2 -√2/2  -1/2   0

This table is the bridge between the unit circle and the graph: each row becomes plotted points.

Routine step 2: Plot points on the θ–y plane

For y = sin θ, plot (θ, sin θ) for each key angle. For y = cos θ, plot (θ, cos θ). Make sure the horizontal axis is labeled in radians and includes 0 through 2π.

Routine step 3: Draw a smooth curve (don’t connect with straight segments)

Sine and cosine change smoothly as the point moves smoothly around the circle. After plotting, draw a smooth wave passing through all points. Use the known peaks and zeros to guide the shape:

• Sine: starts at 0, rises to 1 at π/2, returns to 0 at π, falls to −1 at 3π/2, returns to 0 at 2π.
• Cosine: starts at 1, falls to 0 at π/2, reaches −1 at π, rises to 0 at 3π/2, returns to 1 at 2π.

Routine step 4: Check symmetry and reasonableness

Symmetry helps you catch plotting mistakes.

• Cosine is even: cos(−θ) = cos θ. On a graph, this means symmetry about the y-axis if you extend to negative angles.
• Sine is odd: sin(−θ) = −sin θ. On a graph, this means rotational symmetry about the origin if you extend to negative angles.

Within 0 to 2π, you can still use symmetry ideas by comparing points reflected across π or shifted by π:

• sin(π − θ) = sin θ (same height in Quadrants I and II)
• cos(π − θ) = −cos θ (opposite x-values in Quadrants I and II)
• sin(θ + π) = −sin θ and cos(θ + π) = −cos θ (half-turn flips signs)

Intercepts, key points, and what they mean on the graphs

Intercepts (zeros)

Zeros occur when the corresponding coordinate is 0.

• sin θ = 0 at θ = 0, π, 2π (and every multiple of π)
• cos θ = 0 at θ = π/2, 3π/2 (and every odd multiple of π/2)

Maxima and minima

• sin θ has maximum 1 at θ = π/2 and minimum −1 at θ = 3π/2.
• cos θ has maximum 1 at θ = 0 and 2π, and minimum −1 at θ = π.

Quarter-cycle landmarks

Over one period 0 to 2π, both graphs hit “landmark” values every π/2. These are the easiest checkpoints to memorize and use:

• Sine: 0 → 1 → 0 → −1 → 0
• Cosine: 1 → 0 → −1 → 0 → 1

Short comparison tasks (use the unit circle, not a calculator)

For each, decide which value is larger and justify by thinking about where the angle is on the unit circle (and whether sine/cosine is increasing or decreasing in that region).

• Which is larger: sin(2π/3) or sin(π/6)?
• Which is larger: cos(5π/6) or cos(3π/4)?
• Which is larger: sin(7π/6) or sin(11π/6)?
• Which is larger: cos(π/3) or cos(2π/3)?
• Put in order from least to greatest: sin(3π/2), sin(π), sin(π/2).

Mini-quiz: reading sine and cosine values from graphs

Assume each graph is one full period from 0 to 2π with the usual vertical scale from −1 to 1. Answer using exact values (0, ±1/2, ±√2/2, ±√3/2, ±1) when appropriate.

Part A: Sine graph questions

• At θ = 0, what is sin θ?
• At θ = π/2, what is sin θ?
• At θ = π, what is sin θ?
• At θ = 3π/2, what is sin θ?
• At θ = 2π, what is sin θ?
• If the sine graph crosses y = 1/2 in Quadrant I, which key angle is that?
• If the sine graph is at y = −√2/2, give one key angle in (π, 2π) where this occurs.

Part B: Cosine graph questions

• At θ = 0, what is cos θ?
• At θ = π/2, what is cos θ?
• At θ = π, what is cos θ?
• At θ = 3π/2, what is cos θ?
• At θ = 2π, what is cos θ?
• If the cosine graph is at y = 1/2 in Quadrant I, which key angle is that?
• If the cosine graph is at y = −√3/2 in Quadrant II, which key angle is that?

Part C: Identify the function from key features

• A wave starts at y = 1 when θ = 0, crosses y = 0 at θ = π/2, and reaches y = −1 at θ = π. Is it sine or cosine?
• A wave starts at y = 0 when θ = 0 and reaches y = 1 at θ = π/2. Is it sine or cosine?