1.4 - The Tangent Function and Cofunctions

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Objectives:

  • Understand the relationship between the tangent function and sine and cosine.
  • Understand the relationship between tangent and right triangles.
  • Be able to use the definitions of reciprocal trigonometric functions.
  • Determine if two functions are cofunctions.
Definitions
tangent, secant, cosecant, cotangent, cofunction

Lesson Guide

The Tangent Function

Start by introducing the tangent function. There are, again, two different definitions. You can introduce them at the same time, or put some examples in between the two definitions. 

 Given an angle $\theta$ (in either degrees or radians), we define $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$.

Or, we may define tangent as follows:



$\tan(\theta)=\frac{opposite}{adjacent}=\frac{o}{a}$. This is the "TOA" part of the mnemonic SOH-CAH-TOA, and should be read as "tangent is opposite over adjacent." Again, these two definitions agree, since $\frac{\sin(\theta)}{\cos(\theta)}=\frac{o/h}{a/h}=\frac{o}{a}$.

The Graph of Tangent

Have students calculate the value of $\tan(\theta)$ for each of the angles on their units circles. As a class, use these values to sketch a graph of the tangent function (again, inputs are in radians). Point out to students that the graph of $\tan(\theta)$, despite being periodic, does not have a midline or amplitude. This is because it is not a wave-like periodic function.

Have students work through Problems 1-2 on their worksheets.

Reciprocal Trig Functions & Cofunctions

 There are three other commonly used trigonometric functions, defined below: 
 Secant: $\sec(\theta) = \frac{1}{\cos(\theta)}$
 Cosecant: $\csc(\theta) = \frac{1}{\sin(\theta)}$
 Cotangent: $\cot(\theta) = \frac{1}{\tan(\theta)}$
  • Two functions are called \underline{cofunctions} if they are equal on complementary angles (i.e., angles adding to $90^\circ$, or equivalently $\pi/2$ radians).
  • Sine and cosine are examples of cofunctions (hence the ``co in ``cosine).

You may illustrate this relationship as follows: note that if two angles are complementary, they may be the acute angles of a right triangle. For instance, in the diagram below, $\phi+\theta=90^{\circ}$. Using our SOH-CAH definitions, $\sin(\theta)=a/c$, and $\cos(\phi)=a/c$.


Have students complete Problems 3 and 4 on their worksheets. You may want to remind students what even and odd functions are before doing Problem 4. 

 Just as sine has cofunction cosine, secant and tangent also have cofunctions: 
 $\sin(\theta)=\cos\left(\frac{\pi}{2}-\theta\right)$,
 $\sec(\theta)= \csc\left(\frac{\pi}{2}-\theta\right)$, and 
 $\tan(\theta)=\cot\left(\frac{\pi}{2}-\theta\right).$

Have students complete the remainder of the worksheet.

Comments

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