Preview Activity 4.2.1.
Through the following questions, we work to understand the special values and overall behavior of the tangent function.
(a)
Without using computational device, find the exact value of \(\tan(t)\) at the following values: \(t = \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{2\pi}{3}, \frac{3\pi}{4}, \frac{5\pi}{6}\text{.}\)
(b)
Why is \(\tan \left( \frac{\pi}{2} \right)\) not defined? What are three other input values \(x\) for which \(\tan(x)\) is not defined?
(c)
Point your browser to http://gvsu.edu/s/0yO (“zero-y-Oh”) to find a Desmos worksheet with data from the tangent function already input, denoted in Desmos by the function \(T(x) = \tan(x)\text{.}\) Click on several of the orange points to compare your exact values in (a) with the decimal values given by Desmos. Add one entry to the table: \(x = \frac{11\pi}{24}\text{,}\) \(y = T(\frac{11\pi}{24})\text{.}\) At about what coordinates does this point lie? What are the approximate values of \(\sin(\frac{11\pi}{24})\) and \(\cos(\frac{11\pi}{24})\text{?}\) Why is the value of \(\tan(\frac{11\pi}{24})\) so large relative to the other values of \(\tan(x)\) in the table?
(d)
At the top of the input lists on the left side of the Desmos worksheet, click the circle to highlight the function \(T(x) = \tan(x)\) and thus show its plot along with the data points in orange. Use the plot and your work above to answer the following important questions about the tangent function:
-
What is the domain of \(y = \tan(x)\text{?}\)
-
What is the period of \(y = \tan(x)\text{?}\)
-
What is the range of \(y = \tan(x)\text{?}\)
