Flip an Angle Negative or Subtract It from 90° — Why Sine and Cosine Trade Places: Negative and Complementary Angles

Introduction

Study trigonometry for a while and a flood of rules pours out — rules that flip signs and swap names.

Here \(\theta\) (theta) is the Greek letter used for an angle. Many people memorize these four lines whole: "put in a negative and sine flips sign while cosine stays," and "subtract from \(90°\) and sine and cosine swap places." But ask why only sine flips sign, or why subtracting from \(90°\) swaps the names, and the memorized answer runs dry.

These rules are not a list to memorize. Watch just how a single point on the unit circle moves, and all four lines fall out on their own. Making the angle negative and subtracting it from \(90°\) both come down to one thing: which mirror you reflect the point in. In this post we uncover the identity of those two mirrors.

One piece of terminology first. When we measure an angle, the ray with one end fixed at the origin and rotated by the given angle is called the terminal side. Like the second hand of a clock, a needle starting from an axis and turned by an angle \(\theta\) is the terminal side, and the point where its tip meets the circle of radius \(1\) (the unit circle) we will call \(P\). That point's horizontal coordinate is the cosine, its vertical coordinate the sine.

Hold on to this one sentence and the picture handles the rest.

1. Making the angle negative — the horizontal-axis mirror

What does it mean to hand in a negative angle? Growing the angle positively turns the terminal side counterclockwise. So a negative angle means turning the opposite way — clockwise. If \(P\) is where you land turning counterclockwise by \(\theta\), then \(-\theta\) is where you land turning clockwise by the same amount — that is, \(P\) flipped down across the horizontal (x-)axis.

Think of a reflection. Treat the horizontal axis as the surface of water and reflect \(P\) into it; that watery reflection is exactly the point for \(-\theta\). Watch how the coordinates change under this flip.

• The horizontal coordinate stays the same. We flipped only up and down, so the left–right position does not budge.
• The vertical coordinate only changes sign. A height above the water becomes an equal depth below it — same size, opposite direction (sign).

The horizontal coordinate was cosine, the vertical one sine. So translating this flip straight into the language of coordinates:

Nothing to memorize. The horizontal-axis mirror flips only the vertical part (sine) and leaves the horizontal part (cosine) untouched — that one sentence is both formulas. The reason only sine flips sign is that sine is precisely the vertical coordinate, the direction being flipped.

2. Why the names "odd function" and "even function"

This property has proper names. A function like cosine, whose value is unchanged when you feed it \(-\theta\), is called an even function; a function like sine, whose sign flips when you feed it \(-\theta\), is called an odd function. An even function is symmetric left–right about the vertical axis; an odd function overlaps itself when rotated \(180°\) about the origin.

The names come from powers. With an even exponent like \(x^2\), feeding in \(-x\) lets the minus signs pair up and vanish, so the value is unchanged — "even." With an odd exponent like \(x^3\), one minus sign is left over and the value flips — "odd." Cosine inherits the even-power behavior, sine the odd-power behavior. (Where this even/odd symmetry comes from is covered in more detail in Why Even and Odd Functions Are "Even" and "Odd".)

What about tangent? Tangent was sine divided by cosine. The numerator (sine) flips sign and the denominator (cosine) stays, so the quotient flips sign.

So tangent is an odd function, just like sine. With nothing new to memorize, the negative-angle formulas for all three functions flow out of the single horizontal-axis mirror rule.

3. Subtracting from 90° — the diagonal mirror

Now the second transformation. What happens when you subtract the angle from \(90°\), that is, feed in \(90° - \theta\)? This time it is not the sign but the names of sine and cosine themselves that swap.

The friendliest picture is a right triangle. Its three angles sum to \(180°\), and since one of them is the right angle (\(90°\)), the remaining two acute angles sum to exactly \(90°\). So if one acute angle is \(\theta\), the other is automatically \(90° - \theta\). Two angles that sum to \(90°\) like this are called complementary angles.

Now, the "height" (the opposite side) as seen from one acute angle becomes the "base" (the adjacent side) as seen from the other. Turn the triangle to look from the other angle and the roles of height and base swap outright. Sine was the ratio of height to hypotenuse and cosine the ratio of base to hypotenuse, so switching your viewing angle to the complement makes sine and cosine inherit each other's roles. Hence "the sine of an angle equals the cosine of its complement."

On the unit circle this swap is even sharper. Plot the point \(P\) for angle \(\theta\) and the point \(Q\) for angle \(90° - \theta\) side by side, and \(Q\) sits where \(P\) has been flipped across the diagonal (the line \(y = x\)). The diagonal mirror is the mirror that swaps horizontal and vertical wholesale — instead of reflecting into water, it folds left–right and up–down together along the diagonal. So \(P\)'s horizontal coordinate (cosine) becomes \(Q\)'s vertical coordinate (sine), and \(P\)'s vertical coordinate (sine) becomes \(Q\)'s horizontal coordinate (cosine). The coordinates trade places wholesale, and with them the names sine and cosine.

In fact, the "co" in cosine comes from exactly this complementary relationship. Cosine originally meant "complementary sine." Cosine is not some other function than sine — it is simply sine seen from the complementary angle.

4. Two mirrors in one picture

Both transformations turned out to be one story: how you change the angle decides which mirror the point is reflected in.

• Negative angle \(-\theta\) → reflect the point in the horizontal axis → only the vertical part (sine) flips sign → cosine stays, sine and tangent reverse sign.
• Complementary angle \(90° - \theta\) → reflect the point in the diagonal → horizontal and vertical swap wholesale → sine ↔ cosine trade names.

The key is that a trig function is, in the end, a coordinate of a point on a circle. How you transform the angle fixes which mirror that point lands in, and how the coordinates change is the formula. Instead of memorizing a whole table of sign rules, you only ask, "which mirror is this transformation?" Make the angle negative → horizontal-axis mirror; subtract from \(90°\) → diagonal mirror; the coordinates tell you the rest.

5. See it for yourself

In the interactive below, move the \(\theta\) slider. The buttons at the top let you switch between the two mirrors.

In negative mode, the point \(P\) for angle \(\theta\) (blue) and the point for angle \(-\theta\) (red) face each other like mirror images across the horizontal axis. The green bars marking their horizontal coordinate (cosine) are always the same length and sit together, while only the bars for the vertical coordinate (sine) flip up and down in sign.

In complementary mode, the point \(P\) for angle \(\theta\) and the point \(Q\) for angle \(90° - \theta\) face each other across the diagonal. The length of \(P\)'s horizontal bar exactly matches \(Q\)'s vertical bar, and conversely \(P\)'s vertical matches \(Q\)'s horizontal, so you can watch sine and cosine trade places with your own eyes. Check too that the values on the number panel below match all the way down to the decimals.

Key takeaways

• A trig function is, in the end, a coordinate of a point on the unit circle (horizontal = cosine, vertical = sine). Transforming the angle fixes which mirror the point lands in, and the change in coordinates is the formula.
• The negative angle \(-\theta\) is the horizontal-axis mirror. Only the vertical part (sine) flips sign; the horizontal part (cosine) stays: \(\cos(-\theta)=\cos\theta\) (even function), \(\sin(-\theta)=-\sin\theta\) (odd function), \(\tan(-\theta)=-\tan\theta\).
• The complementary angle \(90°-\theta\) is the diagonal mirror. Horizontal and vertical swap wholesale, so sine and cosine trade places: \(\sin(90°-\theta)=\cos\theta\), \(\cos(90°-\theta)=\sin\theta\). The "co" in cosine comes straight from "complementary."
• Instead of memorizing a table of sign and name rules, just ask "which mirror is this transformation?"

The rules for flipping and subtracting angles are not separate things to memorize but pictures of the same point reflected in different mirrors. One point going around a circle and two mirrors — that was the true identity of a whole page of formulas.

Related reading — A Right Triangle Only Reaches 90° — So How Do We Measure sin 120°? (redefining the ratios as coordinates on a circle — the starting point of this post) · Why Even and Odd Functions Are "Even" and "Odd" (the root of negative-angle symmetry) · Why the Sine, Cosine and Tangent Graphs Look the Way They Do (why cosine is sine pushed sideways)