One Point, Two Trigonometric Values
A point rotating around a circle has a horizontal position and a vertical position. On a circle of radius \(1\) centered at the origin, those two coordinates give a direct geometric meaning to cosine and sine: cosine records the horizontal coordinate, while sine records the vertical coordinate.
This interpretation makes the signs and relative sizes of trigonometric values visible. A point to the left of the vertical axis has a negative horizontal coordinate, for example, so its cosine is negative. A point higher on the circle has a greater vertical coordinate, so its sine is greater.
The angle is measured from the positive horizontal axis, and positive rotation is counterclockwise. The unit circle has radius \(1\), so the point’s coordinates describe its position on that circle. The ordered pair lists the horizontal coordinate first and the vertical coordinate second.
Why the Coordinates Are Cosine and Sine
In the first quadrant, draw a vertical line from \(P(\theta)\) to the horizontal axis. This forms a right triangle. Its hypotenuse is a radius of the unit circle, so its length is \(1\). The horizontal leg has length \(x\), and the vertical leg has length \(y\). By the right-triangle definitions, cosine is adjacent side divided by hypotenuse, and sine is opposite side divided by hypotenuse. Since the hypotenuse is \(1\), these ratios equal the coordinate lengths.
This argument directly explains the coordinate interpretation in the first quadrant. In other quadrants, the same horizontal and vertical positions are used, but a coordinate may be negative. Cosine and sine are signed coordinates, not always positive side lengths. The point’s location determines each sign.
For instance, in Quadrant II the point is left of the vertical axis and above the horizontal axis. Therefore its horizontal coordinate, and hence its cosine, is negative; its vertical coordinate, and hence its sine, is positive. The corresponding sign patterns in the other quadrants follow from whether the point lies left or right of the vertical axis and above or below the horizontal axis.
Reading the Coordinates
To find \(\cos\theta\), read the point’s horizontal coordinate. To find \(\sin\theta\), read its vertical coordinate. The order matters: the first entry of \(P(\theta)\) is cosine, and the second is sine.
| Point’s position | Horizontal coordinate | Vertical coordinate |
|---|---|---|
| Right of the vertical axis | Positive cosine | Depends on whether the point is above or below the axis |
| Left of the vertical axis | Negative cosine | Depends on whether the point is above or below the axis |
| Above the horizontal axis | Depends on whether the point is left or right of the axis | Positive sine |
| Below the horizontal axis | Depends on whether the point is left or right of the axis | Negative sine |
Because the point stays on a circle of radius \(1\), neither coordinate can lie beyond \(1\) or below \(-1\). Thus both cosine and sine values are between \(-1\) and \(1\), inclusive. At the four points on the coordinate axes, one coordinate is \(0\); the other is \(1\) or \(-1\).
Worked Example: Read Both Values in Quadrant II
Worked Example: Read Both Values in Quadrant II
Find \(\cos(5\pi/6)\) and \(\sin(5\pi/6)\) by identifying the point’s coordinates.
The angle \(5\pi/6\) lies in Quadrant II. Its reference angle is \(\pi/6\), whose first-quadrant point has coordinates \((\sqrt{3}/2,1/2)\). Reflecting that point across the vertical axis changes the sign of the horizontal coordinate and leaves the vertical coordinate unchanged. Therefore,
The first coordinate is cosine, and the second is sine. Hence
Check: the point is left of the vertical axis and above the horizontal axis, as required for Quadrant II. Its cosine is negative and its sine is positive, matching the coordinates.
Comparing Values by Comparing Positions
Two cosine values can be compared by comparing the horizontal positions of their points. The point farther to the right has the greater cosine, because its horizontal coordinate is greater. Two sine values can be compared by comparing vertical positions: the point higher up has the greater sine.
This comparison concerns one coordinate at a time. Two different points may share the same horizontal coordinate, giving equal cosines, or share the same vertical coordinate, giving equal sines. A reflection across an axis can make such relationships especially clear: reflecting across the vertical axis reverses the horizontal coordinate while preserving the vertical one.
Worked Example: Compare Two Cosines
Determine which is greater: \(\cos(\pi/3)\) or \(\cos(2\pi/3)\).
The point at \(\pi/3\) is in Quadrant I. Using the exact unit-circle coordinates for this special angle gives
The point at \(2\pi/3\) is its reflection across the vertical axis. Its horizontal coordinate changes sign, so
Since \(1/2>-1/2\), it follows that
Check: the first point is to the right of the vertical axis, and the second is to the left. Their horizontal coordinates have the corresponding positive and negative signs, and the point on the right has the greater horizontal coordinate.
Worked Example: Identify Equal Sines from a Reflection
Compare \(\sin(\pi/6)\) and \(\sin(5\pi/6)\).
The first-quadrant point at \(\pi/6\) is
The point at \(5\pi/6\) is the reflection of this point across the vertical axis:
In each ordered pair, the second coordinate is \(1/2\). Therefore
so the two sine values are equal. Check: the reflection changes only the horizontal position. Both points remain at the same height, which is exactly why their vertical coordinates—and their sines—are equal.
A Reliable Coordinate Method
When evaluating or comparing sine and cosine, organize the reasoning around the point rather than trying to remember signs separately. First identify the angle’s location and the relevant unit-circle point. Then read the horizontal coordinate for cosine or the vertical coordinate for sine. If comparing values, compare the appropriate coordinates on the number line.
Determine the point on the unit circle, using a known exact value or a reference angle and quadrant.
Record the horizontal coordinate first and the vertical coordinate second.
The first coordinate is cosine; the second coordinate is sine.
Confirm that the signs agree with the point’s quadrant and that each coordinate lies between \(-1\) and \(1\).
Common Mistake: Switching the Coordinates
A frequent error is to treat cosine as the vertical coordinate and sine as the horizontal coordinate. That reverses the ordered pair and gives the wrong values, even if the two numbers themselves are familiar.
Worked Example: Correct a Coordinate Mix-Up
At \(\theta=\pi/3\), the point is \(P(\pi/3)=(1/2,\sqrt{3}/2)\). A mistaken calculation might say
This is wrong because it assigns the vertical coordinate to cosine and the horizontal coordinate to sine. By definition, the ordered pair is \((\cos\theta,\sin\theta)\), so the corrected values are
Check: at \(\pi/3\), the point is above and to the right of the origin, and it is higher than it is far to the right. Thus its vertical coordinate is \(\sqrt{3}/2\), while its horizontal coordinate is \(1/2\), as in the corrected assignment.
Why the Coordinate View Is Useful
The unit-circle interpretation turns function values into positions. It explains why cosine can be negative even though a triangle side length cannot be negative: cosine is a signed horizontal coordinate. It also makes comparisons geometric. Instead of comparing two expressions without a picture, one can ask which point is farther right or higher.
The interpretation also provides quick checks. A proposed value outside the interval from \(-1\) to \(1\) cannot be a unit-circle coordinate. A sign inconsistent with the point’s quadrant cannot be correct. These checks do not replace careful evaluation, but they often reveal a reversed coordinate or sign error immediately.
Check Your Understanding
Use the unit-circle coordinate interpretation to answer the questions.
- If a point on the unit circle lies in Quadrant III, what are the signs of its cosine and sine?
- At a point \(P(\theta)=(-1/2,\sqrt{3}/2)\), what are \(\cos\theta\) and \(\sin\theta\)?
- Two points have the same vertical coordinate, but different horizontal coordinates. Which trigonometric values must be equal?
- Explain why reflecting a point across the vertical axis preserves its sine but reverses the sign of its cosine.
- A proposed unit-circle value of \(\cos\theta\) is \(1.2\). What geometric fact shows that this value cannot be correct?
