The Six Graphs

Up to now the angle has been a position on a circle. Now put the angle on a horizontal axis instead, running left to right, and plot the value of the function vertically. Every angle gets a point, and the points join into a curve.

That single change turns a rotation into a wave. Going round the circle once corresponds to moving a distance of 2π along the horizontal axis, so the curve does one complete repeat in that distance. Going round again draws the identical shape again, one repeat further along.

This is why the same functions describe both a rotating wheel and a travelling wave. The two pictures are the same object with different axes. Anything that goes round produces a wave when you plot it against time, and the connection is this unwrapping and nothing more.

The horizontal axis is usually marked in radians, at multiples of π over 2. Marking it in degrees is legal and gives an identical picture with different tick labels.

Sine starts at 0. At angle 0 the point is at (1, 0), on the x axis, so its height above the axis is 0. It rises to 1 at 90 degrees, falls back through 0 at 180 degrees, reaches minus 1 at 270 degrees, and returns to 0 at 360 degrees.

Cosine starts at 1 and falls. At angle 0 the point is as far right as it gets, so its x coordinate is at its maximum of 1. It falls through 0 at 90 degrees, to minus 1 at 180 degrees, back through 0 at 270 degrees, and up to 1 at 360 degrees.

The two curves are identical in shape. Cosine is sine shifted to the left by 90 degrees, which is π over 2 radians. That shift is the cofunction fact from page 3 seen in a different picture: cos θ = sin(90 degrees minus θ) says the same thing about a reflection, and cos θ = sin(θ + 90 degrees) says it as a shift. Both are true, because the sine curve is symmetric in the right way for them to agree.

Four measurements describe each curve. The amplitude Half the distance between the minimum and maximum values of a wave, that is, how far it departs from its centre line. Sine and cosine have amplitude 1. Tangent, cotangent, secant and cosecant have no amplitude, because they are unbounded. is half the distance from the lowest point to the highest, which is 1 for both. The 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. is the horizontal distance for one full repeat, which is 2π for both. The domain The set of inputs a function accepts. Sine and cosine accept every real number. Tangent and secant exclude the angles where cosine is zero, and cotangent and cosecant exclude the angles where sine is zero, because a division by zero has no value there. is the set of inputs the function accepts, which for sine and cosine is every real number, because you can rotate by any angle at all. The range The set of outputs a function produces. Sine and cosine have range [-1, 1], capped by the radius of the unit circle. Tangent and cotangent range over every real number. Secant and cosecant take every value of magnitude at least 1 and nothing strictly between -1 and 1. is the set of outputs it produces, which is every value from minus 1 to 1 inclusive, written [-1, 1]. The range is capped because the circle has radius 1 and no point on it is more than 1 from either axis.

Tangent looks nothing like the other two. It rises from minus infinity, passes through 0 at angle 0, and climbs away to infinity as the angle approaches 90 degrees. Then it restarts from the bottom and does it again.

The breaks are at 90 degrees, 270 degrees, and every 180 degrees from there, which is where cosine is 0 and the quotient sin θ over cos θ divides by zero. At those angles the function has no value. The curve approaches a vertical line there, getting closer and closer without ever touching it. A line a curve approaches arbitrarily closely but never reaches is an asymptote A line a curve approaches arbitrarily closely without ever reaching. In these graphs every asymptote is vertical and sits where the definition divides by zero: tangent and secant at the zeros of cosine, cotangent and cosecant at the zeros of sine. The general rule is that a reciprocal function has an asymptote exactly where the original function has a zero., and asymptotes appear in these graphs wherever a definition divides by zero.

The period is π, not 2π. Tangent repeats every half turn rather than every full turn, which is unlike sine and cosine. The reason is on the circle: rotating by 180 degrees negates both coordinates, and tangent is y over x, so a negative over a negative gives the original value back. Both signs flip and the quotient does not notice.

The range The set of outputs a function produces. Sine and cosine have range [-1, 1], capped by the radius of the unit circle. Tangent and cotangent range over every real number. Secant and cosecant take every value of magnitude at least 1 and nothing strictly between -1 and 1. is every real number. There is no cap, because the tangent segment on the tangent line can be any length at all. The domain The set of inputs a function accepts. Sine and cosine accept every real number. Tangent and secant exclude the angles where cosine is zero, and cotangent and cosecant exclude the angles where sine is zero, because a division by zero has no value there. is every real number except the angles where cosine is 0, and those excluded angles are exactly where the asymptotes are.

Cotangent is cos θ over sin θ, so it breaks where sine is 0. Sine is 0 at 0 degrees, 180 degrees, and every 180 degrees from there, so the asymptotes sit at those angles instead.

Its period is π, the same as tangent, and for the same reason: a half turn flips both coordinates and the quotient is unchanged.

Its range is every real number, again like tangent.

The visible difference is the direction. Each piece of the tangent curve rises from bottom left to top right. Each piece of the cotangent curve falls from top left to bottom right. That follows from the cofunction relationship, cot θ = tan(90 degrees minus θ), which reflects the tangent curve horizontally, and reflecting a rising curve gives a falling one.

Cotangent crosses zero where tangent has its asymptotes, at 90 degrees and every 180 degrees after. That is the general reciprocal rule, stated properly in the next step.

Here is the general rule for any reciprocal, and it saves memorising two more graphs.

Where the original function is 0, the reciprocal has an asymptote, because 1 divided by 0 has no value and 1 divided by a very small number is very large. Where the original is 1, the reciprocal is 1. Where the original is minus 1, the reciprocal is minus 1. Where the original is small, the reciprocal is large, and the other way round. And a reciprocal always has the same sign as the original, since 1 divided by a negative number is negative.

Apply that to cosine. Secant has asymptotes wherever cosine crosses zero, which is 90 degrees, 270 degrees and every 180 degrees. Between consecutive asymptotes, secant is a single U shape, sitting directly inside the arch of the cosine curve and touching it at the arch’s peak. When cosine peaks at 1, secant bottoms out at 1. When cosine bottoms at minus 1, secant peaks at minus 1, opening downwards.

Secant’s period is 2π, inherited from cosine. Its range is every value of size at least 1, which is written as everything less than or equal to minus 1 together with everything greater than or equal to 1. It never takes any value strictly between minus 1 and 1, and page 3 explained why: the shortest distance from the origin to the tangent line is 1.

Cosecant is the identical story against sine. Asymptotes at 0 degrees, 180 degrees and every 180 degrees after, period 2π, and the same gap in its range.

Feed in a negative angle, which means rotating clockwise instead of anticlockwise, and compare with the positive one.

A clockwise rotation lands at the mirror image of the anticlockwise one, reflected in the x axis. Reflecting in the x axis leaves the x coordinate alone and flips the sign of the y coordinate. Since cosine is the x coordinate and sine is the y coordinate, that gives cos(minus θ) = cos θ and sin(minus θ) = minus sin θ.

A function with f(minus x) = f(x) is called an even function A function with f(−x) = f(x), whose graph is symmetric about the vertical axis. Cosine and secant are even. The name comes from powers: x², x⁴ and x⁶ have this property, and the power series for cosine contains only even powers.. Its graph is symmetric about the vertical axis: fold the paper along that axis and the two halves land on each other. Cosine is even, and so is secant, since a reciprocal keeps the sign.

A function with f(minus x) = minus f(x) is called an odd function A function with f(−x) = −f(x), whose graph has rotational symmetry about the origin. Sine, cosecant, tangent and cotangent are odd. The name comes from powers: x, x³ and x⁵ have this property, and the power series for sine contains only odd powers.. Its graph has rotational symmetry about the origin: turn the paper 180 degrees about the origin and the curve lands on itself. Sine is odd, and so are cosecant, tangent and cotangent. Tangent is odd because it is an odd function divided by an even one.

The names come from powers. x², x⁴ and x⁶ are even functions, x, x³ and x⁵ are odd functions, and the parity of the exponent is where the words come from. The connection is not decorative: the infinite series for cosine contains only even powers of x, and the series for sine only odd ones.