FFR (For Future Reference)

Back to trigonometry

This section is somewhat unique. At the end of each chapter, there will be a section of important concepts to keep in mind, called “For Future Reference” (FFR). Some of these concepts you may already know, or should know. Others may be additive on the existing information.

In future FFR sections, a range of miscellaneous topics that do not deserve their own chapter will be covered. Since Mathematics builds on itself, it is important to review concepts that will be used in future sections. Here are a list of things you should already know, but are reviewed briefly.

Function Domains and Ranges

Recall that the domain of a function is made up of all possible x-values you can assign the function without the function being undefined.

The range of a function is made up of all possible values the function can return.

Square brackets vs parentheses: In order to represent the minimum possible range, square brackets ([ ]) are used. When representing a range that the function APPROACHES, parentheses are used as mathematical convention (( )).

Some tips:

• Any value or range of values of x that makes the function undefined is not included in the domain. For example, logarithmic functions have no negative numbers or 0 in their domain.
• Similarly, if a function cannot equal a certain value, no matter what domain (x-value) you give it, then that set of values is not within the function’s range. For example, e^x has no negative numbers or zeroes in its range because an exponential function can never be negative.

Example Problem 1.3.4

What is the domain and range of the function $3\cdot \sin (\ln (x)$ ?

Log Rules

The next important review topic that should be touched upon is logarithm rules. Although you learned about logarithms way back in Algebra 2, some of you might be rusty, like I was when I started Calculus. Listed below are the most common log laws you will use. The most common bases are base 10 and base e (represented by ln). In computer programming, other bases may be used such as base 2 and base 16.

Chapter 1 Problems

1. Calculate ${\left(\text{csc}\left(\frac{5\pi }{4}\right)•\text{tan}\left(\frac{5\pi }{4}\right)\right)}^2-1$. Refer to the unit circle.
2. Calculate $\frac{\text{cot}\left(\frac{\pi }{6}\right)\text{csc}\left(\frac{\pi }{6}\right)}{\left(\text{sec}\left(\frac{\pi }{8}\right)+\text{tan}\left(\frac{\pi }{8}\right)\right)•\left(\text{sec}\left(\frac{\pi }{8}\right)-\text{tan}\left(\frac{\pi }{8}\right)\right)}$

3. Calculate ${\left(\text{csc}\left(\frac{3\pi }{4}\right)•\text{tan}\left(\frac{3\pi }{4}\right)\right)}^2+{\left(\text{sec}\left(\frac{3\pi }{4}\right)•\text{cot}\left(\frac{3\pi }{4}\right)\right)}^2$
4. Write the equation for the graph below with sin(x) as the base function (Hint: Notice the maxima and minima).

5. Write the equation for the graph in number 4 with cos(x) as the base function rather than sin(x).
6. If the function f(x) = $e^x$ were shifted one unit to the right, one unit vertically up, and vertically compressed by $\frac{1}{2}$ , what would be the new function?

   Clarification: Apply the vertical compression before the upward translation; equivalently use the combined transform g(x)=a f(x−h)+k.

7. What is the domain and range of the function f(x) = ${\text{sin}}^{-1}(x)$ ? Refer to the unit circle.
8. Convert the equation $y={\text{log}}_4(20)+{\text{log}}_{10}(16)$ into the form y = ln(c). Then find the value of y. You may want to refer to log rules.

9. Simplify $y=\text{ln}{\left(\frac{3e}{\pi }\right)}^2$ such that the argument of the ln function only has one term (ex: $y=\text{A ln}(a)+\text{B ln}(b)+\text{C ln}(c)$. Refer to the log rules
10. (You knew this one was coming) Given that the black graph is y = $e^x$ , write the equations of all of the other colored graphs.

    Clarification: Identify the standard exponential/logarithmic reflections that match the labeled raster graph. The equations are inferred from the graph.