The Unit Circle

A right triangle has one right angle, so its other two angles are both smaller than 90 degrees. There is no right triangle with a 120 degree angle in it. There is no right triangle with a negative angle in it either.

So the definitions from page 1 cover angles from 0 to 90 degrees and nothing else. That is a real problem, because the things trigonometry is used for do not stop at 90 degrees. A wheel turns through 360 degrees and keeps going. A pendulum swings to both sides of vertical. An alternating current is negative for half of every cycle.

There are two ways out. Declare that sin(120 degrees) has no meaning, or find a definition that agrees with the triangle for acute angles and keeps working outside that range. The second is what mathematics did, and the tool is a circle.

The extension is not arbitrary. There are many ways to extend a function beyond where it was defined, and most of them are useless. The circle definition is chosen because it agrees exactly with the old one on the old range, and because the resulting functions repeat, which is the behaviour that turning and oscillating things actually have.

Draw a circle centred at the origin The point (0, 0) where the coordinate axes cross, and the centre of the unit circle. Angles are measured from the positive x axis at this point, and the rotating radius has one end fixed here., the point (0, 0) where the two axes cross. Give it a radius The distance from the centre of a circle to any point on it, and also the line segment joining them. On the unit circle it is 1, which is what makes the sine and cosine of an angle equal to the coordinates of the point rather than merely proportional to them. of exactly 1. The word unit The circle of radius 1 centred at the origin, with equation x² + y² = 1. A radius drawn at angle theta from the positive x axis meets it at the point (cos θ, sin θ), which is taken as the definition of sine and cosine for every angle. It agrees with the right triangle definitions for angles between 0 and 90 degrees and extends them to obtuse, reflex and negative angles. means one, and that is the whole meaning of the name: the unit circle The circle of radius 1 centred at the origin, with equation x² + y² = 1. A radius drawn at angle theta from the positive x axis meets it at the point (cos θ, sin θ), which is taken as the definition of sine and cosine for every angle. It agrees with the right triangle definitions for angles between 0 and 90 degrees and extends them to obtuse, reflex and negative angles. is the circle of radius 1 centred at the origin. There are no units attached, no centimetres or inches. It is 1.

Now put a line from the centre out to the rim, and let it rotate. Angles are measured from the positive x axis, the horizontal line pointing right, and turning anticlockwise counts as positive. That convention is a choice, and it is the one everyone uses.

The triangle is what sits underneath that rotating radius. Drop a vertical line from the point where the radius meets the circle, straight down to the x axis. You now have a right triangle: the radius is the hypotenuse, the vertical drop is the opposite side, and the run along the x axis is the adjacent side.

The radius does not change length as it turns. That is what a circle is. So this triangle always has a hypotenuse of exactly 1, whatever the angle, and that constant is what the next step exploits.

Apply the triangle definitions to this triangle. Sine is opposite over hypotenuse, and the hypotenuse is 1, so sin θ is the opposite side divided by 1, which is the opposite side. Cosine is adjacent over hypotenuse, so cos θ is the adjacent side.

The opposite side is the height of the point above the x axis, which is its y coordinate. The adjacent side is the distance along the x axis, which is its x coordinate. So the point where the rotating radius meets the circle has coordinates (cos θ, sin θ). Not proportional to them. Equal to them.

That is the new definition. Cosine of an angle is the x coordinate of the corresponding point on the unit circle. Sine is the y coordinate. No triangle required.

Because no triangle is required, the definition keeps working when a triangle would be impossible. Rotate to 120 degrees and the point sits up and to the left, with a negative x coordinate, so cos(120 degrees) is negative. Rotate to 200 degrees and both coordinates are negative. Rotate backwards, to a negative angle, and the point drops below the axis. Every one of these is a legitimate point on a circle, so every one has coordinates, so every one has a sine and a cosine.

This is a redefinition, and it is worth being clear about what a redefinition has to prove. It has to agree with the old definition everywhere the old one applied. It does: for any angle between 0 and 90 degrees the point is in the upper right, the triangle underneath it is a genuine right triangle, and the two definitions give the same number. Outside that range the old definition said nothing, so there is nothing to contradict.

Degrees are arbitrary. A full turn is 360 degrees because 360 divides neatly by a lot of numbers and because Babylonian arithmetic ran in base 60. Nothing about a circle produces the number 360.

A radian The angle that cuts an arc equal in length to the radius. A full turn is 2π radians (about 6.283) and π radians is 180 degrees, so one radian is 180/π degrees, about 57.296. Radians are the standard unit in analysis because on the unit circle the arc length equals the angle, which removes a conversion factor from every formula built on top, including the derivatives of sine and cosine. is defined from the circle itself. Take the radius, bend it round the rim, and mark where the end lands. The angle at the centre between the start and that mark is one radian. So one radian is the angle that cuts an arc equal in length to the radius.

The circumference of a circle of radius r is 2πr, which is 2π radius lengths laid end to end. So a full turn is exactly 2π radians, which to three decimal places is 6.283 radians. Half a turn is π radians, and half a turn is 180 degrees, so π radians equals 180 degrees. Dividing gives one radian equal to 180 divided by π degrees, which is 57.296 degrees to three decimal places.

The reason to bother is that radians make the arc length equal to the angle. On the unit circle, an angle of θ radians cuts an arc of exactly θ units. In degrees the same statement needs a conversion factor of π over 180 stuck in front of it. That factor then propagates into every formula built on top, including the derivatives of sine and cosine, which come out clean in radians and cluttered in degrees.

Both scales describe the same angles. Degrees are convenient for talking; radians are what the mathematics is written in. The word radian is recent: James Thomson used it in an examination paper in 1873.

The triangle under the radius has legs of length equal to the x and y coordinates of the point, and a hypotenuse of 1. Apply Pythagoras: x² + y² = 1².

That is the equation of the unit circle. Every point on the circle satisfies it, and every point that satisfies it lies on the circle. The equation of a circle is not a separate fact from Pythagoras’ theorem. It is Pythagoras’ theorem with the hypotenuse named as a radius.

Now substitute what the coordinates are. x is cos θ and y is sin θ. So (cos θ)² + (sin θ)² = 1. The standard way to write that is cos²θ + sin²θ = 1, where the exponent sits after the function name to avoid confusion with an inverse function.

That equation holds for every angle, without exception. Not for particular angles you have to look up. Every one. An equation that is true for every value of its variable is called an identity An equation that holds for every value of its variable, as opposed to an equation solved for particular values. sin²θ + cos²θ = 1 is an identity; sin θ = 0.5 is an equation. The Pythagorean, reciprocal, quotient and cofunction identities are the four families that connect the six functions., and this is the first and most important identity in the subject. Page 6 derives the other two Pythagorean identities from it by division, and neither of them is an independent fact.

One immediate consequence: since a square is never negative and the two squares add to 1, neither sin θ nor cos θ can ever be outside the interval from minus 1 to 1. The circle has radius 1, so no point on it is further than 1 from the origin in any direction.

Rotate by a full turn and you are back where you started. The point on the rim is the same point, so its coordinates are the same numbers, so the sine and cosine are the same values.

In symbols, sin(θ + 2π) = sin θ and cos(θ + 2π) = cos θ, for every θ. In degrees that is a repeat every 360 degrees. A function that repeats its values at a fixed interval like this is called periodic Describing a function whose values repeat at a fixed interval, so f(x + p) = f(x) for every x. All six trigonometric functions are periodic because rotating by a whole number of full turns returns to the same point on the circle., and the smallest interval over which it repeats is its period The smallest interval over which a periodic function repeats. Sine, cosine, secant and cosecant have period 2π (360 degrees). Tangent and cotangent have period π (180 degrees), because a half turn negates both coordinates and their quotient is unchanged.. Sine and cosine have period 2π.

Negative angles work the same way. A negative angle means rotate clockwise instead of anticlockwise. An angle of minus 90 degrees puts the point at the bottom of the circle, at coordinates (0, minus 1), so cos(minus 90 degrees) is 0 and sin(minus 90 degrees) is minus 1. Rotating clockwise by 90 degrees and anticlockwise by 270 degrees end at the same place, and therefore give identical values.

This is why these functions describe anything that repeats. A wheel, a wave, a vibrating string, an alternating current, a planet in orbit. The repetition is not a side effect of the definition, it is the defining behaviour of a circle, imported wholesale.

One consequence for solving equations. If you ask which angle has a sine of 0.5, there is no single answer, because adding any whole number of full turns gives another. Page 5 handles this properly.