Reciprocal functions and properties of trigonometric graphs

The three reciprocal functions

• Each is one divided by one of the familiar functions:

$$\sec\theta = \frac{1}{\cos\theta} \qquad \csc\theta = \frac{1}{\sin\theta} \qquad \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$$

• The pairing is deliberately counterintuitive:
  • sec goes with cos (not sin)
  • cosec goes with sin (not cos)
  • cot goes with tan
• The memory aid: look at the third letter. Sec → cos; cosec → sin... more reliably, remember that the "co" in cosec pairs it with the function without a "co".

Where they are undefined

• A reciprocal is undefined wherever its partner is zero.
  • $\sec\theta$ is undefined where $\cos\theta = 0$, at $\theta = \frac{\pi}{2} + k\pi$
  • $\csc\theta$ is undefined where $\sin\theta = 0$, at $\theta = k\pi$
  • $\cot\theta$ is undefined where $\sin\theta = 0$, at $\theta = k\pi$
• These are vertical asymptotes on the graphs, and they must be marked on any sketch.

Their ranges

• Since $|\sin\theta| \leq 1$ and $|\cos\theta| \leq 1$, taking reciprocals gives:

$$|\sec\theta| \geq 1 \qquad |\csc\theta| \geq 1$$

• So $\sec\theta$ and $\csc\theta$ never take values strictly between $-1$ and $1$. An equation such as $\sec\theta = 0.5$ has no solution, and recognising that immediately saves wasted work.
• $\cot\theta$ has no such restriction — its range is all real numbers, like $\tan\theta$.

The shape of the basic graphs

Function	Period	Range	Asymptotes
$\sin\theta$	$2\pi$	$-1 \leq y \leq 1$	none
$\cos\theta$	$2\pi$	$-1 \leq y \leq 1$	none
$\tan\theta$	$\pi$	all real $y$	$\theta = \frac{\pi}{2} + k\pi$
$\sec\theta$	$2\pi$	$\lvert y \rvert \geq 1$	$\theta = \frac{\pi}{2} + k\pi$
$\csc\theta$	$2\pi$	$\lvert y \rvert \geq 1$	$\theta = k\pi$
$\cot\theta$	$\pi$	all real $y$	$\theta = k\pi$

• Note that tangent and cotangent have period $\pi$, not $2\pi$. This matters enormously when solving equations — a tangent equation has solutions spaced $\pi$ apart, not $2\pi$.

Transformations

• For $y = a\sin\left(b(\theta - c)\right) + d$:

  • $|a|$ — the amplitude, half the distance from minimum to maximum
  • $b$ — affects the period: new period $= \dfrac{2\pi}{b}$
  • $c$ — the horizontal shift (phase shift), right by $c$
  • $d$ — the vertical shift, up by $d$

• The period formula is the one to be careful with. $y = \sin 3\theta$ has period $\frac{2\pi}{3}$ — a larger $b$ means a shorter period, because the graph is squashed horizontally.

• For tangent, the period is $\dfrac{\pi}{b}$, since the parent period is $\pi$.

• Worked through — $y = 4\cos\left(2\theta\right) - 1$:

  • Amplitude 4, so the curve swings 4 either side of its centre line
  • Period $\frac{2\pi}{2} = \pi$
  • Centre line at $y = -1$, so the range is $-5 \leq y \leq 3$

Symmetry properties

• Even and odd behaviour is worth knowing because it appears in identity work:

$$\cos(-\theta) = \cos\theta \qquad \text{(even — symmetric about the } y\text{-axis)}$$

$$\sin(-\theta) = -\sin\theta \qquad \text{(odd — rotationally symmetric about the origin)}$$

$$\tan(-\theta) = -\tan\theta \qquad \text{(odd)}$$

• Shift relationships, which explain why sine and cosine look identical apart from position:

$$\sin\left(\theta + \frac{\pi}{2}\right) = \cos\theta \qquad \cos\left(\theta - \frac{\pi}{2}\right) = \sin\theta$$

Modelling with trigonometric functions

• Anything periodic can be modelled this way — tides, daylight hours, temperature cycles, a Ferris wheel.
• Build the model from the features:
  • Amplitude $= \dfrac{\text{max} - \text{min}}{2}$
  • Vertical shift $= \dfrac{\text{max} + \text{min}}{2}$ — the centre line
  • $b$ from the period: $b = \dfrac{2\pi}{\text{period}}$
  • $c$ from where the cycle starts