These are the tasks you should be able to perform with reasonable fluency when you arrive at our next class meeting. Important new vocabulary words are indicated in italics.
State the basic properties of all six main trigonometric functions: definition, domain, graph, values of the function at the main angles (\(0,\pi/6,\pi/4,\pi/3,\pi/2\) and all geometrically related angles), related trigonometric identities.
📝 [Submit] Explore the applet Sine, Cosine, and Tangent animated from the unit circle. Before starting this section, it is important to remember how the tangent function is related to the sine and cosine functions. Use this applet to answer the following questions. Submit screen captures and answers to the questions posed as needed.
Check the \(\sin \theta\) box. As you move the slider for the angle \(\theta\text{,}\) the point \(P\) on the unit circle moves and the graph of \(\sin \theta\) is plotted. How is the height of the blue point on the graph above \(\theta\) related to point \(P\text{?}\) Hint: Look at the blue dashed line in the unit circle.
Now check the \(\cos \theta\) box. As you move the slider for the angle \(\theta\text{,}\) the point \(P\) on the unit circle moves and the graph of \(\cos \theta\) is plotted. How is the height of the red point on the graph above \(\theta\) related to point \(P\text{?}\) Hint: Look at the red dashed line in the unit circle.
Now check the \(\tan \theta\) box. As you move the slider for the angle \(\theta\text{,}\) the point \(P\) on the unit circle moves and the graph of \(\tan \theta\) is plotted. How is the height of the purple point on the graph above \(\theta\) related to point \(P\text{?}\) Hint: Look at the red and blue dashed lines in the unit circle.
Explore the applet located at https://www.geogebra.org/m/hFReSaqJ. This applet gives you a geometric way to think about the value of \(\tan \beta\) when \(0^{\circ}\leq \beta \leq 90^{\circ}\text{.}\) As you change the angle (via the slider), it strongly suggests that the length of \(\overline{TB}\) is \(\tan \beta\text{.}\) Knowing that \(\tan \beta\) is the ratio of lengths of \(\overline{DP}\) to \(\overline{OD}\) where \(O\) is the origin, how could we prove that indeed the length of \(\overline{TB}\) is equal to \(\tan \beta\text{?}\)
Based on the graph of \(f(x) = \tan (x)\text{,}\) on a single period \((-\pi,\pi)\text{,}\) what can you say about concavity? What does this suggest about the graph of \(f'(x)\) on \((-\pi,\pi)\text{?}\)
Prompt Copilot “I want a pneumonic device to remember the derivatives of the 6 basic trigonometric functions.” Is the device it returns useful to you? Or more confusing?
Derive the derivatives of \(y=\tan x, y=\sec x, y=\cot x, and y=\csc x\) “from scratch” using only basic derivative rules and trigonometric identities.
Use all rules learned so far in the context of a real-world problem to find the slope of a tangent line, the instantaneous rate of change in a function, or the instantaneous velocity of an object.
