The Six Trigonometric Functions — An Interactive Chapter

Transformations

Every graph in this chapter is drawn from one general form,

\[ y = A\, f\bigl(B(x - C)\bigr) + D, \]

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.

The general form, with the current values

y = 1 · f( 1(x − 0) ) + 0

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.

Angle \(\theta\)

45°•0.785 rad•π/4

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 \(\sin\theta = 0\).

Functions in the overlay

Select which functions are drawn together in §2.

§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.

\[ \sin\theta = y, \qquad \cos\theta = x, \qquad \tan\theta = \frac{y}{x} = \frac{\sin\theta}{\cos\theta}, \qquad \cot\theta = \frac{x}{y} = \frac{\cos\theta}{\sin\theta}, \qquad \sec\theta = \frac{1}{x} = \frac{1}{\cos\theta}, \qquad \csc\theta = \frac{1}{y} = \frac{1}{\sin\theta}. \]

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

\[ \sin^2\theta + \cos^2\theta = 1. \]

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 \(\pi\).

§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.

§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 \(|\csc\theta|\).

\(\cos\theta\) — the \(x\)-coordinate of \(P\)

\(\sin\theta\) — the \(y\)-coordinate of \(P\)

\(\tan\theta\) — the segment on \(x = 1\), from \((1, 0)\) to \(T\)

\(\cot\theta\) — the segment on \(y = 1\), from \((0, 1)\) to \(S\)

\(\sec\theta\) — the segment \(OT\), where \(T = (1, \tan\theta)\) \(\sec\theta =\) —

\(\csc\theta\) — the segment \(OS\) (dashed), where \(S = (\cot\theta, 1)\) \(\csc\theta =\) —

the angle \(\theta\) and the point \(P(\cos\theta, \sin\theta)\)

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 \(\sin\theta = 0\).

§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.