THE TRIGONOMETRIC FUNCTIONS

Up to now in this Precalculus course, only two trigonometric functions have been studied extensively—sine and cosine.
There's a reason for this!
All the remaining trigonometric functions are defined in terms of sine and cosine.

This section discusses the four remaining trigonometric functions: tangent, cotangent, secant, and cosecant.

Definitions: Tangent, Cotangent, Secant, Cosecant

DEFINITIONS tangent, cotangent, secant, cosecant
Let $\,t\,$ be a real number (restrictions are noted individually below).
Think of $\,t\,$, if desired, as the radian measure of an angle.
Then:
  • $\displaystyle \tan t := \frac{\sin t}{\cos t}$     (‘$\,\tan t\,$’ is formally pronounced ‘tangent of $\,t\,$’)

    The tangent function isn't defined where the cosine is zero; this happens at the terminal points $\,(0,1)\,$ and $\,(0,-1)\,$.
    Thus, tangent is not defined for $\displaystyle \,t = \frac{k\pi}{2}\,$ for odd integers $\,k\,$:   $\,k = \ldots, -5, -3, -1, 1, 3, 5,\, \ldots\,$
  • $\displaystyle \cot t := \frac{\cos t}{\sin t}$     (‘$\,\cot t\,$’ is formally pronounced ‘cotangent of $\,t\,$’)

    The cotangent function isn't defined where the sine is zero; this happens at the terminal points $\,(1,0)\,$ and $\,(-1,0)\,$.
    Thus, cotangent is not defined for $\displaystyle \,t = k\pi\,$ for integers $\,k\,$:   $\,k = \ldots, -3, -2, -1, 0, 1, 2, 3,\, \ldots\,$
  • $\displaystyle \sec t := \frac{1}{\cos t}$     (‘$\,\sec t\,$’ is formally pronounced ‘secant (see-cant) of $\,t\,$’)

    The secant function has the same restrictions as the tangent.
  • $\displaystyle \csc t := \frac{1}{\sin t}$     (‘$\,\csc t\,$’ is formally pronounced ‘cosecant of $\,t\,$’)

    The cosecant function has the same restrictions as the cotangent.

Notes on the Definitions:

Memory Device for the Six Trigonometric Functions

Cosecant is the reciprocal of the sine.
Secant is the reciprocal of the cosine.
Cotangent is the reciprocal of the tangent.

How might you remember these relationships?
The memory device at right should help!

Write the trig functions vertically, with function followed immediately by cofunction:
sine,   cosine,     tangent,   cotangent,     secant, cosecant

Then, connect the functions as shown—they're reciprocals!

Where Do the Names ‘Tangent’ and ‘Secant’ Come From?

Consider an acute angle $\,\theta\,$ laid off in the unit circle.
Its terminal point is shown in black below.

Only one auxiliary triangle (in yellow below—it overlaps the green triangle)
is needed to explain the significance of the names ‘tangent’ and ‘secant’, as follows:

The green and (overlapping) yellow triangles below are similar.
Why? They share the angle $\,\theta\,$ and they both have a right angle.
The word ‘tangent’ derives from the Latin tangens, meaning ‘touching’.

The tangent gives the length of a segment
that is tangent to the unit circle.



 bottom of triangleright side of triangle
green triangle:$\cos\theta$$\sin\theta$
yellow triangle:$1$$y$
(initially unknown)

By similarity: $$ \frac{\sin\theta}{\cos\theta} = \frac{y}{1} $$ Therefore: $$\color{blue}{y = \frac{\sin\theta}{\cos\theta} := \tan\theta}$$
The word ‘secant’ derives from the Latin secare, meaning ‘to cut’.

The secant gives the length of a segment
that cuts the circle.



 bottom of trianglehypotenuse
green triangle:$\cos\theta$$1$
yellow triangle:$1$$h$
(initially unknown)

By similarity: $$ \frac{1}{\cos\theta} = \frac{h}{1} $$ Therefore: $$\color{blue}{h = \frac{1}{\cos\theta} := \sec\theta}$$

What is the ‘co’ in COsine, COtangent, and COsecant?

In English, to complement (not compliment!) is to fill out or to complete.

By definition, the complement of an angle $\,\theta\,$ (given in degree measure) is $\,90^\circ - \theta\,$.
Similarly, the complement of an angle $\,\theta\,$ (given in radian measure) is $\,\frac{\pi}2 - \theta\,$.
Thus, an (acute) angle and its complement, together, fill out a right angle.

The ‘co’ in cosine, cotangent, and cosecant refers to ‘complement’.
The following relationships (when they're defined) are true for all real numbers $\,\theta\,$ (not just those in the first quadrant):

Here are sketches/explanations to illustrate these relationships in the first quadrant:


By definition: $$\cos\theta = \ell$$ Using the right triangle definition: $$\displaystyle\sin(\frac{\pi}2 - \theta) = \frac{\text{OPP}}{\text{HYP}} = \frac{\ell}{1} = \ell$$ Comparing: $$\cos\theta = \sin(\frac{\pi}{2} - \theta)$$

By definition: $$ \begin{gather} \cos\theta = \ell\cr \sin\theta = m\cr \cot\theta := \frac{\cos\theta}{\sin\theta} = \frac{\ell}{m} \end{gather} $$ Using the right triangle definition: $$\displaystyle\tan(\frac{\pi}2 - \theta) = \frac{\text{OPP}}{\text{ADJ}} = \frac{\ell}{m}$$ Comparing: $$\cot\theta = \tan(\frac{\pi}{2} - \theta)$$

By definition: $$ \begin{gather} \sin\theta = m\cr \csc\theta := \frac{1}{\sin\theta} = \frac{1}{m} \end{gather} $$ Using the definition of secant
and the right triangle definition of cosine: $$\displaystyle\sec(\frac{\pi}2 - \theta) := \frac{1}{\cos(\frac{\pi}2 - \theta)} = \frac{1}{m/1} = \frac{1}{m}$$ Comparing: $$\csc\theta = \sec(\frac{\pi}{2} - \theta)$$

Here are two excellent references for historical and cultural information on the trigonometric functions:

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
signs of all the trigonometric functions
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7 8 9 10 11 12
13 14 15 16 17 18 19 20 21 22 23  
AVAILABLE MASTERED IN PROGRESS

(MAX is 23; there are 23 different problem types.)