📈 Level 2 - Topic 1: Graphing Trigonometric Functions (Sine, Cosine, Tangent) 📊📐

1) Expanding Our Graphing Toolkit - Sine, Cosine, and Tangent 📊

Welcome to Level 2 Trigonometry! In Level 1, you were introduced to the basic graphs of sine and cosine functions. Now, we'll expand our graphing toolkit to include the tangent function and explore the graphs of all three fundamental trigonometric functions – sine, cosine, and tangent – in more detail. Understanding these graphs is crucial for further study in trigonometry and its applications.

Building on Level 1:

Review of Sine and Cosine Graphs: We'll start by briefly recapping the key features of sine and cosine graphs learned in Level 1 (shape, period, amplitude, range).
Introducing the Tangent Graph: We'll then introduce the graph of the tangent function \( y = \tan(\theta) \), which has unique characteristics compared to sine and cosine.
Comparative Analysis: We'll compare and contrast the graphs of sine, cosine, and tangent, highlighting their similarities and differences.
Foundation for Transformations: This topic sets the stage for understanding transformations of trigonometric graphs (phase shift, amplitude, period changes) in the next topic.

2) Review: The Graphs of Sine and Cosine - Key Features Refresher 🔄

Let's quickly recap the essential features of the sine and cosine graphs we learned about in Level 1. This will help us build a foundation for comparing them with the tangent graph.

Sine Function: \( y = \sin(\theta) \)

Shape: Sinusoidal Wave

Period: \( 360^\circ \) or \( 2\pi \) radians

Amplitude: 1

Range: \( [-1, 1] \)

Passes through origin: Yes, \( \sin(0) = 0 \)

Cosine Function: \( y = \cos(\theta) \)

Shape: Sinusoidal Wave

Period: \( 360^\circ \) or \( 2\pi \) radians

Amplitude: 1

Range: \( [-1, 1] \)

Starts at Maximum: Yes, \( \cos(0) = 1 \)

3) Introducing the Graph of the Tangent Function - \( y = \tan(\theta) \) 🚧

Now, let's turn our attention to the graph of the tangent function, \( y = \tan(\theta) \). The tangent graph behaves quite differently from sine and cosine graphs. Remember that \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). This definition is key to understanding its graph.

Understanding Tangent from Sine and Cosine:

When \( \sin(\theta) = 0 \), \( \tan(\theta) = 0 \) (as long as \( \cos(\theta) \neq 0 \)). Tangent graph crosses the \( \theta \)-axis when sine graph does (at multiples of \( 180^\circ \) or \( \pi \) radians).
When \( \cos(\theta) = 0 \), \( \tan(\theta) \) is undefined because division by zero is not allowed. This happens at \( \theta = 90^\circ, 270^\circ, -90^\circ, \) etc. ( \( \frac{\pi}{2} + n\pi \) radians). At these points, the tangent graph has vertical asymptotes.
When \( \cos(\theta) \) is close to zero and \( \sin(\theta) \) is not, \( \tan(\theta) \) becomes very large (positive or negative).

Key Features of the Tangent Graph \( y = \tan(\theta) \):

Shape: Repeating curves that approach vertical lines (asymptotes). Not a wave like sine or cosine.
Period: \( 180^\circ \) or \( \pi \) radians. The graph repeats its pattern every \( 180^\circ \) (or \( \pi \)). Shorter period than sine and cosine.
Amplitude: No Amplitude. The tangent function does not have a maximum or minimum value in the same way as sine and cosine. Its range is unbounded.
Range: \( (-\infty, \infty) \) or all real numbers. Tangent values can be any real number.
Vertical Asymptotes: Occur at \( \theta = 90^\circ + n \cdot 180^\circ \) or \( \theta = \frac{\pi}{2} + n\pi \) (where \( n \) is any integer). The graph approaches these vertical lines but never touches them.
Passes through origin: Yes, \( \tan(0) = 0 \).

4) Comparing Sine, Cosine, and Tangent Graphs - Key Differences Summarized 📊🔍

Let's directly compare the graphs of \( y = \sin(\theta) \), \( y = \cos(\theta) \), and \( y = \tan(\theta) \) to highlight their key differences.

Comparative Summary: Sine, Cosine, and Tangent Graphs

Feature	Sine \( y = \sin(\theta) \)	Cosine \( y = \cos(\theta) \)	Tangent \( y = \tan(\theta) \)
Shape	Sinusoidal Wave	Sinusoidal Wave	Repeating Curves with Asymptotes
Period	\( 360^\circ \) or \( 2\pi \)	\( 360^\circ \) or \( 2\pi \)	\( 180^\circ \) or \( \pi \)
Amplitude	1	1	None
Range	\( [-1, 1] \)	\( [-1, 1] \)	\( (-\infty, \infty) \)
Vertical Asymptotes	None	None	Yes, at \( \theta = 90^\circ + n \cdot 180^\circ \)
Crosses \( \theta \)-axis	At \( \theta = n \cdot 180^\circ \)	At \( \theta = 90^\circ + n \cdot 180^\circ \)	At \( \theta = n \cdot 180^\circ \)
Starts at \( \theta = 0^\circ \)	\( y = 0 \)	\( y = 1 \) (Maximum)	\( y = 0 \)

5) Practice Questions 🎯

5.1 Fundamental – Graph Identification & Feature Recognition

1. Which of the three graphs (sine, cosine, tangent) has vertical asymptotes? Describe where these asymptotes occur.

2. Which of the three graphs has a period of \( 180^\circ \) (or \( \pi \) radians)?

3. Which of the three graphs has an amplitude? What is the amplitude for those that have it?

4. Which of the three graphs has a range of all real numbers \( (-\infty, \infty) \)?

5. For what angles in the range \( 0^\circ \leq \theta < 360^\circ \) does the tangent graph cross the \( \theta \)-axis (y = 0)?

6. True or False: Both sine and cosine graphs are bounded, while the tangent graph is unbounded. Explain.

7. True or False: The sine and cosine graphs are horizontal shifts of each other, while the tangent graph has a fundamentally different shape. Explain.

8. Match each function (sine, cosine, tangent) with its starting y-value at \( \theta = 0^\circ \): (a) \( y = 0 \), (b) \( y = 1 \), (c) Undefined.

9. What is the smallest positive angle at which the tangent function has a vertical asymptote?

10. Sketch a basic cycle of the tangent graph, indicating the positions of the vertical asymptotes and where it crosses the \( \theta \)-axis.

5.2 Challenging – Conceptual & Comparative Questions 💪🚀

1. Explain why the tangent function has vertical asymptotes at \( \theta = 90^\circ + n \cdot 180^\circ \). Relate your explanation to the definitions of sine and cosine at these angles.

2. Compare and contrast the periodic behavior of sine and cosine graphs with the periodic behavior of the tangent graph. What is the significance of the period in each case?

3. Why is it meaningful to talk about the "amplitude" of sine and cosine graphs, but not for the tangent graph? How does the range of each function relate to this?

4. Consider the behavior of the tangent function as \( \theta \) approaches \( 90^\circ \) from the left and from the right. Explain what happens to \( \tan(\theta) \) and how this is reflected in the graph near the asymptote.

5. (Conceptual) Imagine you are walking along the \( \theta \)-axis from \( 0^\circ \) to \( 360^\circ \). Describe how your vertical position (y-value) would change if you were walking along the sine graph, the cosine graph, and then the tangent graph. Highlight the key differences in your "walks" along each graph.

6) Summary - Graphing Sine, Cosine, and Tangent 🎉

Graphing Toolkit Expanded: Level 2 starts with graphing all three: sine, cosine, and tangent.
Sine and Cosine Review: Sinusoidal waves, period \( 360^\circ \) (\( 2\pi \)), amplitude 1, range \( [-1, 1] \). Cosine is a phase-shifted sine.
Tangent Graph \( y = \tan(\theta) \): Repeating curves, period \( 180^\circ \) (\( \pi \)), no amplitude, range \( (-\infty, \infty) \), vertical asymptotes at \( \theta = 90^\circ + n \cdot 180^\circ \).
Key Differences: Tangent has shorter period, unbounded range, and vertical asymptotes, unlike sine and cosine.
Comparative Understanding: Understanding similarities and differences of these graphs is crucial for further trigonometry.

Excellent work! You now have a solid understanding of the graphs of sine, cosine, and tangent functions, and how they compare. Understanding these fundamental graphs is essential for exploring more advanced trigonometric concepts and applications in Level 2. Get ready to move on to Topic 2, where we'll learn about transformations of these graphs - phase shift, amplitude changes, and period adjustments! 📈📊📐🌟