The Unit Circle: Coordinates as Cosine and Sine

Defining the Unit Circle

The unit circle is the circle centered at the origin with radius 1. In coordinate form, it is the set of all points (x, y) that satisfy x² + y² = 1.

To connect angles to points, we place an angle θ in standard position: its vertex at the origin and its initial side on the positive x-axis. The terminal side intersects the unit circle at exactly one point (x, y).

Why (x, y) = (cos θ, sin θ)

On the unit circle, the radius is 1, so the point where the terminal side hits the circle has a special meaning: x is the cosine of the angle and y is the sine of the angle. We write:

(x, y) = (cos θ, sin θ)

This is not a new formula to memorize so much as a new interpretation: every angle θ corresponds to a point on the unit circle, and the coordinates of that point are the cosine and sine values.

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Step-by-step: Reading sine and cosine from a point

• Find the point (x, y) on the unit circle for angle θ.
• Read cos θ as the x-coordinate.
• Read sin θ as the y-coordinate.

Example: If the point is (0.6, 0.8), then cos θ = 0.6 and sin θ = 0.8 (and it should satisfy 0.6² + 0.8² = 1).

Incremental Visuals: Start with Key Angles

Begin by anchoring a few “compass point” angles. These are easy because the terminal side lands on an axis.

• θ = 0: point (1, 0) so (cos θ, sin θ) = (1, 0)
• θ = π/2: point (0, 1) so (0, 1)
• θ = π: point (-1, 0) so (-1, 0)
• θ = 3π/2: point (0, -1) so (0, -1)

These four points divide the circle into four quadrants and give you immediate sign information for sine and cosine.

Activity: Label the axis points

Draw a circle of radius 1 centered at the origin. Mark the four axis intersection points and label each with its coordinate pair. Next to each point, write the matching angle (0, π/2, π, 3π/2).

Quadrants and Signs of Sine and Cosine

Quadrants tell you the signs of x and y, which immediately tells you the signs of cosine and sine.

• Quadrant I (x > 0, y > 0): cos θ positive, sin θ positive
• Quadrant II (x < 0, y > 0): cos θ negative, sin θ positive
• Quadrant III (x < 0, y < 0): cos θ negative, sin θ negative
• Quadrant IV (x > 0, y < 0): cos θ positive, sin θ negative

Step-by-step: Determine signs from quadrant alone

• Identify the quadrant where the terminal side lies.
• Use the quadrant sign pattern for x and y.
• Translate x-sign to cos-sign and y-sign to sin-sign.

Example: If θ is in Quadrant II, then cos θ is negative and sin θ is positive, even before you know any exact values.

Activity: Sign-only practice

For each description, write the sign of cos θ and sin θ (use + or -):

• θ is in Quadrant III
• θ is in Quadrant I
• θ is in Quadrant IV
• θ is in Quadrant II

Using Symmetry to Generate Coordinates

Once you know a point (cos θ, sin θ) in one location, you can get related points by reflecting across axes or rotating by π. This is powerful because many angles share the same coordinate magnitudes, with signs changing predictably.

Reflection rules (same magnitudes, sign changes)

Suppose a point on the unit circle is (x, y) = (cos θ, sin θ).

• Reflect across the x-axis: (x, -y). Cosine stays the same; sine changes sign.
• Reflect across the y-axis: (-x, y). Sine stays the same; cosine changes sign.
• Rotate by π (180°) about the origin: (-x, -y). Both sine and cosine change sign.

Step-by-step: Generating related angles from one known point

• Start with a known point (x, y) for some angle θ.
• Choose a symmetry move (reflect across x-axis, reflect across y-axis, rotate by π).
• Apply the coordinate sign change to get the new point.
• Use the new point to read cos and sin for the related angle.

Example: If an angle in Quadrant I has point (x, y), then the corresponding point reflected across the y-axis is (-x, y), which lies in Quadrant II. That means the related angle has cosine negative and sine positive, with the same magnitudes as before.

Incremental Visuals: From One Special Angle to Many

Pick a single “reference” point in Quadrant I at a special angle (for example, θ = π/6, π/4, or π/3). You already know the coordinate magnitudes for these angles from special triangles; on the unit circle, those become the x- and y-coordinates.

Now generate the other three points with the same magnitudes:

• Quadrant II: (-x, y)
• Quadrant III: (-x, -y)
• Quadrant IV: (x, -y)

This creates a set of four angles that share the same absolute values of cosine and sine, differing only by signs.

Activity: Build a “symmetry square” of coordinates

Choose one Quadrant I coordinate pair (x, y) for a special angle. Fill in the other three quadrants using symmetry:

• QI: (x, y)
• QII: (____, ____)
• QIII: (____, ____)
• QIV: (____, ____)

Then, for each quadrant, write the sign of cos and sin without computing anything else.

Identifying an Angle from a Point

Sometimes you are given a point on the unit circle and asked to identify the angle(s). The process is: (1) determine the quadrant from signs, (2) match the coordinate magnitudes to a known special-angle pair, and (3) choose the angle in that quadrant.

Step-by-step: From point to angle

• Check the signs of x and y to find the quadrant.
• Use |x| and |y| to recognize the special-angle coordinate magnitudes.
• Select the angle in that quadrant that has those cosine and sine values.

Example pattern: If you recognize |x| and |y| as the magnitudes from a common special angle in Quadrant I, then the angle in another quadrant will have the same magnitudes but with the quadrant’s sign pattern.

Mini-Quiz: Unit Circle Coordinates and Signs

Part A: Read coordinates as (cos θ, sin θ)

• 1) If the point is (0, -1), what are cos θ and sin θ?
• 2) If the point is (-1, 0), what are cos θ and sin θ?
• 3) If the point is (x, y) with x > 0 and y < 0, what are the signs of cos θ and sin θ?

Part B: Identify quadrant and signs

• 4) θ is in Quadrant II. State the sign of cos θ and the sign of sin θ.
• 5) θ is in Quadrant III. State the sign of cos θ and the sign of sin θ.
• 6) θ is in Quadrant I. State the sign of cos θ and the sign of sin θ.

Part C: Symmetry moves

• 7) A point on the unit circle is (x, y). What is the reflected point across the x-axis?
• 8) A point on the unit circle is (x, y). What is the reflected point across the y-axis?
• 9) A point on the unit circle is (x, y). What is the point after a rotation by π?

Part D: Angles from axis points

• 10) Which angle corresponds to the point (1, 0)?
• 11) Which angle corresponds to the point (0, 1)?
• 12) Which angle corresponds to the point (0, -1)?