§1 Definitions
Let \(\theta\) be an angle measured from the positive \(x\)-axis, and let \(P = (x, y)\) be the point where the terminal side of \(\theta\) meets the unit circle — the circle of radius \(1\) centred at the origin. The six trigonometric functions of \(\theta\) are defined from the coordinates of \(P\) as follows.
Two of the six — sine and cosine — are the coordinates themselves. Tangent and cotangent are ratios of the coordinates, and secant and cosecant are reciprocals. Every fact in this chapter follows from these definitions together with the equation of the circle, \(x^2 + y^2 = 1\), which gives the fundamental identity
Because the point \(P\) returns to its starting position after a full turn, all six functions are periodic: their values repeat. Sine, cosine, secant, and cosecant have period \(2\pi\); tangent and cotangent, which depend only on the ratio of the coordinates, repeat after half a turn and have period \(\pi\).
For \(\theta = \frac{\pi}{6}\) the terminal side meets the circle at \(P = \bigl(\frac{\sqrt3}{2}, \frac12\bigr)\): the height of the point is exactly \(\frac12\), as the \(30^\circ\!-\!60^\circ\!-\!90^\circ\) triangle in the diagram shows. The definitions now give every value by substitution — first the two coordinates, then the two ratios and the two reciprocals:
Nothing else is ever needed: read off \(x\) and \(y\), then form the ratios and reciprocals.
The terminal side now lies in the second quadrant, at \(P = \bigl(-\frac{\sqrt3}{2}, \frac12\bigr)\): the same height as in Example 1.1, but a negative \(x\)-coordinate. Substitution gives
Sine and cosecant, which involve only \(y\), are unchanged; every function involving \(x\) has changed sign. The signs are not extra facts to memorise — they are read off the position of \(P\).
Transformations
An example angle
Choose one angle \(\theta\) to examine in detail. The dashed line \(x = \theta\) meets each graph at the point \(\bigl(\theta, f(\theta)\bigr)\), and the same angle is drawn on the unit circle in §3. The preset angles are the standard angles, whose exact values are known in closed form.
Functions in the overlay
Select which functions are drawn together in the graph below.
§2 The graphs
\(x \in [-2\pi,\, 2\pi]\)The graph of a function \(f\) is the set of points \(\bigl(x, f(x)\bigr)\): as \(x\) runs through two full periods, the height of the graph at each \(x\) is the value defined in §1 for that angle. Drawing all six on one set of axes makes their relationships visible. Cosine is sine shifted left by \(\frac{\pi}{2}\). Each reciprocal function mirrors its base function — it is large exactly where the base function is near zero, crosses \(\pm 1\) where the base function crosses \(\pm 1\), and has no value where the base function is zero. Tangent and cotangent increase or decrease through every real value once in each period. The resulting breaks in the four non-waveform graphs are analysed in §5.
Values at \(\theta\)
| Function | Value | Exact value |
|---|
The middle column gives the value of the transformed function at the example angle, as a decimal. The last column gives the exact value of the basic function, and is shown only when the transformations are switched off (\(A = 1\), \(B = 1\), \(C = 0\), \(D = 0\)) and \(\theta\) is a standard angle.
A value is undefined exactly when a denominator in its definition is zero: tangent and secant where \(\cos\theta = 0\), cosecant and cotangent where \(\sin\theta = 0\).
The four transformations
Every graph in this chapter is drawn from one general form,
where \(f\) is any one of the six functions. The four constants each perform one geometric transformation of the basic graph. They are applied to all six functions at once, so their effects can be compared directly.
Amplitude \(A\). A vertical stretch. Every output is multiplied by \(A\), so for sine and cosine the amplitude — half the distance between the maximum and minimum values — becomes \(|A|\) instead of \(1\). Tangent and the reciprocal functions have no maximum or minimum, but \(A\) scales each of their values in the same way.
Frequency \(B\). A horizontal compression. The input reaches each stage of its cycle \(B\) times as fast, so one complete cycle occupies \(\frac{2\pi}{|B|}\) instead of \(2\pi\) for sine, cosine, secant, and cosecant, and \(\frac{\pi}{|B|}\) instead of \(\pi\) for tangent and cotangent. Doubling \(B\) doubles the number of cycles in any fixed interval.
Phase shift \(C\). A horizontal shift. Replacing \(x\) by \(x - C\) moves every point of the graph to the right by \(C\): the value formerly attained at \(x\) is now attained at \(x + C\). Note the sign convention — \(x - C\) shifts right, while \(x + C\) would shift left.
Vertical shift \(D\). A vertical translation. Adding \(D\) moves the whole graph up by \(D\). Sine and cosine then oscillate about the horizontal line \(y = D\), called the midline, instead of about the \(x\)-axis.
An example angle
Choose the angle \(\theta\) drawn in the diagram below. The preset angles are the standard angles, whose exact values are known in closed form.
§3 The unit circle
θ = 45°Each function can also be read from the diagram as the length of a line segment. Construct the diagram in four steps. (1) Draw the angle \(\theta\) and mark the point \(P\) where its terminal side meets the circle. (2) The projections of \(P\) onto the axes are its coordinates: the horizontal segment has length \(|\cos\theta|\) and the vertical segment \(|\sin\theta|\). (3) Draw the two tangent lines \(x = 1\) and \(y = 1\). The segment cut off on \(x = 1\) by the \(x\)-axis and the line \(OP\) has length \(|\tan\theta|\); the segment cut off on \(y = 1\) has length \(|\cot\theta|\). (4) The same two points of intersection, joined to the centre, give the remaining functions as hypotenuses: if \(T\) is the point on \(x = 1\) and \(S\) the point on \(y = 1\), then the distance from \(O\) to \(T\) is \(|\sec\theta|\) and from \(O\) to \(S\) is \(|\csc\theta|\).
Signs. A segment's length gives the absolute value; the sign comes from the definitions in §1. Sine has the sign of the \(y\)-coordinate and cosine the sign of the \(x\)-coordinate, so secant shares the sign of cosine and cosecant the sign of sine. On the diagram this appears as position: when \(\cos\theta < 0\), for instance, the point \(T\) lies on the extension of the line \(OP\) behind the centre, and \(\sec\theta\) is negative. The example angle on the unit circle is the untransformed \(\theta\) of §1; the constants \(A\), \(B\), \(C\), \(D\) transform graphs, not the defining diagram.
Behaviour near the excluded angles. As \(\theta\) approaches \(\frac{\pi}{2}\), the point \(T\) recedes without bound along \(x = 1\): \(\tan\theta\) and \(\sec\theta\) grow beyond every limit, and at \(\frac{\pi}{2}\) itself they have no value, because \(\cos\theta = 0\). Cosecant and cotangent behave the same way near \(0\) and \(\pi\), where \(\sin\theta = 0\).
Scale. The figure is drawn at one fixed scale at every angle: one unit always spans the same length, so a segment's length at one angle can be compared by eye with its length at any other. The frame extends to about five and a half units in each direction, so the whole construction is in view except very near the excluded angles, where \(T\) and \(S\) pass beyond the frame and their segments are cut off at its edge — the diagrammatic counterpart of the values exceeding every bound.
Transformations
An example angle
Choose one angle \(\theta\) to examine in detail. The dashed line \(x = \theta\) meets each graph at the point \(\bigl(\theta, f(\theta)\bigr)\). The preset angles are the standard angles, whose exact values are known in closed form.
§4 The six functions, one at a time
Each graph below is drawn with the same constants \(A\), \(B\), \(C\), \(D\), so that a transformation can be followed from function to function. Read each card with its definition in §1: the period, the values taken, and the angles excluded from the domain are all consequences of that definition. The checkbox on each card overlays the function's reciprocal — sine with cosecant, cosine with secant, tangent with cotangent — drawn dashed in its own colour with the same constants. Overlaying the pair makes the relationship of §5 visible on a single graph: the zeros of one function are the asymptotes of the other, its extrema are the other's vertices, and the two curves cross where both equal \(1\) or both equal \(-1\).
The four transformations
Every graph in this chapter is drawn from one general form,
where \(f\) is any one of the six functions. The four constants each perform one geometric transformation of the basic graph. They are applied to all six functions at once, so their effects can be compared directly.
Amplitude \(A\). A vertical stretch. Every output is multiplied by \(A\), so for sine and cosine the amplitude — half the distance between the maximum and minimum values — becomes \(|A|\) instead of \(1\). Tangent and the reciprocal functions have no maximum or minimum, but \(A\) scales each of their values in the same way.
Frequency \(B\). A horizontal compression. The input reaches each stage of its cycle \(B\) times as fast, so one complete cycle occupies \(\frac{2\pi}{|B|}\) instead of \(2\pi\) for sine, cosine, secant, and cosecant, and \(\frac{\pi}{|B|}\) instead of \(\pi\) for tangent and cotangent. Doubling \(B\) doubles the number of cycles in any fixed interval.
Phase shift \(C\). A horizontal shift. Replacing \(x\) by \(x - C\) moves every point of the graph to the right by \(C\): the value formerly attained at \(x\) is now attained at \(x + C\). Note the sign convention — \(x - C\) shifts right, while \(x + C\) would shift left.
Vertical shift \(D\). A vertical translation. Adding \(D\) moves the whole graph up by \(D\). Sine and cosine then oscillate about the horizontal line \(y = D\), called the midline, instead of about the \(x\)-axis.
§5 The breaks in the graphs
Sine and cosine are waveforms: their graphs are single unbroken curves, because a coordinate of the point \(P\) varies continuously as the angle varies. The other four functions are formed from the coordinates by division, and division introduces two distinct kinds of break. This section explains both: the vertical asymptotes, where a function has no value and its graph separates into branches, and — for secant and cosecant only — the excluded band of values that are never taken at all. In the graphs of §4 the asymptotes are drawn as dashed vertical lines.
5.1 Why the asymptotes occur
Tangent and secant are quotients with denominator \(\cos\theta\); cotangent and cosecant are quotients with denominator \(\sin\theta\):
Consider what happens as the denominator approaches one of its zeros — say \(\cos\theta \to 0\) as \(\theta \to \frac{\pi}{2}\). The numerator does not approach zero at the same time: the identity \(\sin^2\theta + \cos^2\theta = 1\) means sine and cosine never vanish together, so at \(\theta = \frac{\pi}{2}\) the numerator of tangent is \(\sin\theta = 1\), and the numerator of secant is \(1\) throughout. A quotient whose numerator approaches \(\pm 1\) while its denominator approaches \(0\) cannot approach any finite number: its absolute value grows beyond every bound,
The line \(x = \frac{\pi}{2}\) is called a vertical asymptote of both graphs: the curve draws arbitrarily close to it without ever reaching it, because at \(x = \frac{\pi}{2}\) itself the function has no value — division by zero is not a large number, it is no number at all, and the point is simply absent from the domain and from the graph.
An asymptote should not be confused with a hole. A hole is an isolated missing point at which the two-sided limit is finite — the graph closes in on a definite height from both sides and only the single point is absent, so the gap could in principle be filled by assigning that height. Nothing of the sort is possible here. At an asymptote the values do not close in on any height; they leave every bounded range, and (as the next part shows) they usually leave in opposite directions on the two sides. The discontinuity is therefore not removable: the branches on either side belong to the same formula but to genuinely separate pieces of the graph, and no point drawn at the asymptote could join them.
5.2 Which side goes up, and which goes down
The direction of escape on each side is determined by signs. Near an asymptote the numerator is close to \(\pm 1\) and does not change sign, while the denominator passes through zero and does change sign — sine and cosine cross the axis transversally at their zeros, negative on one side and positive on the other. The quotient therefore has opposite signs on the two sides of every asymptote, so one branch escapes to \(+\infty\) and the other to \(-\infty\). Reading the sign of the denominator on each side gives the complete picture:
| Function | No value at | Approached from the left | Approached from the right |
|---|---|---|---|
| \(\tan\theta\) | \(\theta = \frac{\pi}{2} + k\pi\), where \(\cos\theta = 0\) | \(+\infty\) | \(-\infty\) |
| \(\cot\theta\) | \(\theta = k\pi\), where \(\sin\theta = 0\) | \(-\infty\) | \(+\infty\) |
| \(\sec\theta\), at \(\theta = \frac{\pi}{2}\) | \(\theta = \frac{\pi}{2} + k\pi\), where \(\cos\theta = 0\) | \(+\infty\) | \(-\infty\) |
| \(\sec\theta\), at \(\theta = \frac{3\pi}{2}\) | \(-\infty\) | \(+\infty\) | |
| \(\csc\theta\), at \(\theta = 0\) | \(\theta = k\pi\), where \(\sin\theta = 0\) | \(-\infty\) | \(+\infty\) |
| \(\csc\theta\), at \(\theta = \pi\) | \(+\infty\) | \(-\infty\) |
Evaluate \(\tan\theta\) at angles closing in on \(90^\circ\):
| \(\theta\) | \(89^\circ\) | \(89.9^\circ\) | \(89.99^\circ\) | \(90^\circ\) | \(90.01^\circ\) | \(90.1^\circ\) | \(91^\circ\) |
|---|---|---|---|---|---|---|---|
| \(\tan\theta\) | \(57.290\) | \(572.957\) | \(5729.578\) | undefined | \(-5729.578\) | \(-572.957\) | \(-57.290\) |
Each step closes the remaining gap tenfold and multiplies the value roughly tenfold: from the left the values race away to \(+\infty\), and on the right they reappear from \(-\infty\). The diagram shows the two branches on either side of the dashed asymptote \(x = \frac{\pi}{2}\). No point of the graph has \(x\)-coordinate \(\frac{\pi}{2}\) — the column for \(90^\circ\) is empty not because the value is very large, but because there is no value.
Note the difference in pattern. Tangent and cotangent have period \(\pi\), and their asymptotes are spaced \(\pi\) apart, so the pattern at every asymptote is the same as at the one listed. Secant and cosecant have period \(2\pi\) but their asymptotes are still spaced \(\pi\) apart, so consecutive asymptotes alternate: the pattern at \(\frac{3\pi}{2}\) is the reverse of the pattern at \(\frac{\pi}{2}\), and the pattern at \(\pi\) is the reverse of the pattern at \(0\). One full period always contains one asymptote of each orientation.
The same sign analysis explains the shapes between asymptotes. On any interval between two consecutive asymptotes a non-waveform function is continuous — no denominator zero lies inside — so its graph there is one unbroken branch. Tangent's branch rises from \(-\infty\) to \(+\infty\), taking every real value once; cotangent's falls from \(+\infty\) to \(-\infty\). This is also why the four graphs consist of infinitely many separate branches, one per interval between asymptotes, each a translate of the others by a whole number of periods.
5.3 The excluded band of secant and cosecant
Secant and cosecant have a second kind of gap, horizontal rather than vertical. Because \(|\cos\theta| \le 1\), its reciprocal satisfies \(|\sec\theta| \ge 1\), and similarly \(|\csc\theta| \ge 1\). No point of either graph has a height strictly between \(-1\) and \(1\): the open band \(-1 < y < 1\) is excluded — not merely unvisited near some angle, but unattainable at every angle. Where tangent and cotangent pass through all values, secant and cosecant divide into branches that lie entirely in \(y \ge 1\) or entirely in \(y \le -1\).
Sample \(\sec\theta\) at every \(60^\circ\):
with no value at \(\theta = \pm\frac{\pi}{2}\). Every value attained lies in \(y \ge 1\) or \(y \le -1\). In the diagram the strip between the dashed lines \(y = 1\) and \(y = -1\) is shaded: the graph touches the strip's boundary at its vertices \((0, 1)\) and \((\pm\pi, -1)\) and escapes to infinity at the asymptotes \(x = \pm\frac{\pi}{2}\), but never enters the strip.
Each branch lies between two consecutive asymptotes and has a single vertex, the point where it comes closest to the band. A branch of secant attains \(y = 1\) exactly where \(\cos\theta = 1\), or \(y = -1\) exactly where \(\cos\theta = -1\) — that is, at the maxima and minima of its base function — and from the vertex the branch rises away on both sides toward the asymptotes erected at the zeros of cosine. The picture to hold is reciprocal in the literal sense: where the waveform is tallest, its reciprocal is shortest, and where the waveform sinks to zero, its reciprocal escapes to infinity. The zeros of the base function become the asymptotes of the reciprocal, and the extrema of the base function become the vertices of the reciprocal.
5.4 The breaks under the transformations
For the transformed function \(y = A\, f\bigl(B(x - C)\bigr) + D\), the asymptotes stand where the argument reaches a value at which the basic function has none:
Three consequences follow. (i) Only \(B\) and \(C\) move the asymptotes: \(C\) translates the whole pattern, and \(B\) rescales its spacing from \(\pi\) to \(\frac{\pi}{|B|}\). (ii) Neither \(A\) nor \(D\) can create, remove, or relocate an asymptote — multiplying or shifting heights does not change where a denominator vanishes — and with \(A > 0\) the directions of escape in the table of §5.2 are unchanged. (iii) The excluded band of secant and cosecant moves with \(D\) and widens with \(A\): the unattainable values are exactly the open interval \(\bigl(D - |A|,\ D + |A|\bigr)\), and its endpoints \(D - |A|\) and \(D + |A|\) are attained, at the vertices of the branches. As a check, set \(B = 2\) and \(C = 0\) for tangent: the formula places the asymptotes at \(x = \frac{\pi}{4} + \frac{k\pi}{2}\), twice as densely as the basic graph — and that is what the graph in §4 shows.