A Right Triangle Only Reaches 90° — So How Do We Measure sin 120°? Move the Stage from Triangle to Circle and Even the Signs Fall Out

Introduction

Learn the trigonometric ratios on a right triangle and you hit a wall almost at once. A right triangle already has one angle pinned at \(90°\), so the other two, however large, can never pass \(90°\). And yet type \(\sin 120°\) into a calculator and out comes a perfectly good \(0.866\); \(\sin 210°\) and \(\sin(-60°)\) have values too. These are angles that cannot even exist inside a triangle — so where does their sine come from?

The answer is to change the stage the ratios stand on. On the cramped stage of a triangle the ceiling is \(90°\), but move the stage to a circle and the angle can grow as large as you like and even go negative. This post redefines the trigonometric ratios as a point on a terminal side reaching out from the origin (Section 1), shows why that definition naturally embraces angles beyond \(90°\) and negative angles (Section 3), and why the sign rule (which ratios are positive in each quadrant) then follows without any memorizing (Section 2).

1. Redefine the ratios from 'sides of a triangle' to 'coordinates on a circle'

First, one piece of terminology. A terminal side is a ray with one end fixed at the origin, rotated by the given angle. Picture the second hand of a clock — a needle starting from an axis and turned by an angle \(\theta\) (theta, the Greek letter for an angle) is the terminal side. Seeing the angle not as 'a vertex of a triangle' but as 'the direction a needle points' — that shift of view is the whole idea.

Now place a point other than the origin on the terminal side and call it \(P(x, y)\). Let the distance from the origin to this point be \(r\); by the Pythagorean theorem,

Here we redefine the trigonometric ratios not as ratios of sides but as ratios of coordinate to distance.

Is this really the same as the old definition? If \(\theta\) lies between \(0°\) and \(90°\), the point \(P\) sits in the upper right — the first quadrant. Then \(x\) is the base, \(y\) is the height, and \(r\) is the hypotenuse, so the old right triangle is resurrected exactly. Thus \(y/r\) is 'height over hypotenuse', precisely the old \(\sin\theta\). The right triangle definition is just one special slice of the new one.

The decisive change is this: the coordinates \(x\) and \(y\) can be negative. Turn the terminal side to the left and \(x\) goes negative; turn it downward and \(y\) goes negative. A triangle's side length could never be negative, but a coordinate carries direction and so permits a minus sign. It is exactly this one step that lets angles beyond \(90°\) have values.

To keep the arithmetic simple we usually take a circle of radius \(r=1\) (the unit circle). Then the denominator is \(1\) and drops out:

so the coordinates of the point on the unit circle are the cosine and the sine themselves. Feed in an angle and it fixes a point on the circle; that point's horizontal coordinate is the cosine, its vertical coordinate the sine.

2. r is always positive — so the sign is set by x and y alone

With the new definition in hand, one thing students used to memorize whole simply falls out: the sign rule for which ratio is positive in each quadrant.

The key is the distance \(r\). Since \(r\) is the distance from the origin to the point, it is always positive — a distance can never be negative, whatever the direction. So the sign of \(\sin\theta=y/r\) has nothing to do with the denominator \(r\) and follows only the numerator \(y\). Likewise the sign of \(\cos\theta=x/r\) follows \(x\), and the sign of \(\tan\theta=y/x\) follows the combination of the signs of \(x\) and \(y\). In short:

• Upper half (\(y>0\)) gives \(\sin\theta>0\); lower half (\(y<0\)) gives \(\sin\theta<0\).
• Right half (\(x>0\)) gives \(\cos\theta>0\); left half (\(x<0\)) gives \(\cos\theta<0\).
• \(\tan\theta=y/x\) is \(+\) when \(x\) and \(y\) have the same sign, \(-\) when they differ.

Apply this to the four quadrants and the famous rule appears with no picture needed. The first quadrant (\(x>0,\,y>0\)) is all three \(+\); the second (\(x<0,\,y>0\)) has only \(\sin\) positive; the third (\(x<0,\,y<0\)) has \(x\) and \(y\) sharing a sign, so only \(\tan\) is \(+\); the fourth (\(x>0,\,y<0\)) has only \(\cos\) positive. The table usually memorized as 'All–Sin–Tan–Cos' was, all along, nothing but the fact that \(r>0\) leaving you to read the signs of x and y.

Check it on a concrete example. At \(\theta=120°\) the terminal side points to the upper left — the second quadrant. On the unit circle that point's coordinates are \((-0.5,\ 0.866)\). Then

The horizontal coordinate is negative, so cosine is negative; the vertical coordinate is positive, so sine is positive; their signs differ, so tangent is negative — exactly matching the rule that only sine is positive in the second quadrant. The \(\sin 120°\) that could not exist on a triangle at all gains both a value and a sign the moment we read it off the circle. Note too that \(0.866\) is the decimal form of \(\sqrt{3}/2\), so it has the same magnitude as the familiar \(\sin 60°\), keeping the positive sign.

3. Negative angles and beyond 360° — natural once you see rotation

View an angle as the rotation of a terminal side and two things come for free.

First, negative angles. A rotation has a direction. Having agreed that counterclockwise is \(+\), a clockwise turn naturally becomes a negative angle. For instance \(-60°\) is the terminal side turned \(60°\) clockwise, which points in exactly the same direction as \(300°\) turned counterclockwise. So \(\sin(-60°)=\sin 300°\), and that point lies in the fourth quadrant with \(y<0\), giving \(\sin(-60°)=-0.866\). A 'negative angle' made no sense on a triangle, but under rotation it carries the perfectly ordinary meaning of 'turned the other way.'

Second, beyond one full turn. Rotate the terminal side one more full circle (\(360°\)) and it returns to where it was. Same direction means same coordinates, and same coordinates mean same ratio values. So

This is the periodicity of the trigonometric functions — grow the angle without bound and the values simply repeat every full turn. For example \(405°\) is \(360°+45°\), the same position as \(45°\), and its value matches at \(\sin 45°=0.707\). In this way the ratios shed the yoke of 'one angle of a triangle' and rise to being functions that assign a value to every real angle — trigonometric functions.

4. See it for yourself

In the interactive below, sweep the \(\theta\) slider from \(-180°\) to \(540°\). As the terminal side rotates, the point \(P\) moves around the circle and its coordinates \(x\) and \(y\) appear in the panel as cosine and sine. The instant the point crosses into another quadrant, the sign of \(x\) or \(y\) flips, and with it the color of sine, cosine, and tangent (green for positive, red for negative). The radius \(r\) stays fixed at \(1\) throughout, playing no part in the sign, as the panel notes.

Use the 'example angles' buttons for \(120°\) (second quadrant), \(210°\) (third quadrant), \(-60°\) (a negative angle), and \(405°\) (beyond one turn), and the values and signs computed in the text are reproduced exactly. The sine-wave strip at the bottom shows the value flowing on smoothly past \(90°\) and repeating each full turn — the periodicity made visible.

Key takeaways

• Redefine the trigonometric ratios not as ratios of a right triangle's sides, but through a point \(P(x, y)\) on the terminal side rotated by \(\theta\) from the origin, with distance \(r=\sqrt{x^{2}+y^{2}}\). Then \(\sin\theta=y/r\), \(\cos\theta=x/r\), \(\tan\theta=y/x\), and in the first quadrant this is identical to the old right triangle definition.
• The coordinates \(x\) and \(y\) can go negative with direction, while the distance \(r\) is always positive. So the sign of a trigonometric ratio is set solely by the signs of \(x\) and \(y\) — the per-quadrant sign rule (first all \(+\), second only sine, third only tangent, fourth only cosine) is not something to memorize but something that falls out here.
• Seeing an angle as rotation makes clockwise a negative angle (\(\sin(-60°)=\sin 300°\)) and a full-turn difference the same value (\(\sin(\theta+360°)=\sin\theta\)) — the periodicity. In this way the ratios extend into trigonometric functions defined for every real angle.

This single shift of stage, from triangle to circle, removes the \(90°\) ceiling and hands you the sign rule for free. The graphs of the trigonometric functions, and their extension into waves, all start from this definition on the circle.

Related reading — Why Are Trigonometric Ratios the Same Regardless of a Triangle's Size (the ratio of sides that similarity preserves — this post's right triangle definition) · Why Do We Measure Angles with Real Numbers (Radians) (measuring the circle's angle with real numbers) · What Is the Point of Trigonometric Functions Anyway (where the circle definition extends into waves)