What is the primary difference between the starting points of $y = \sin(x)$ and $y = \cos(x)$ at $x = 0$?

The sine function starts at the origin $(0,0)$, which is the midline of the wave. The cosine function starts at its maximum value $(0,1)$ for the basic parent function.

How does the period of $y = \tan(x)$ differ from $y = \sin(x)$?

The period of $y = \tan(x)$ is $180^\circ$ (or $\pi$ radians), whereas the period of $y = \sin(x)$ is $360^\circ$ (or $2\pi$ radians). This means the tangent function repeats twice as frequently.

When comparing $y = \sin(x)$ and $y = \cos(x)$, what horizontal shift would make them identical?

A horizontal shift of $90^\circ$ (or $\pi/2$ radians) will align the two graphs. Specifically, $\sin(x + 90^\circ) = \cos(x)$.

What is a common error when identifying the period of $y = \sin(2x)$?

Students often think the '2' means the period is $720^\circ$. In reality, the coefficient $b$ compresses the graph, making the period $360^\circ / 2 = 180^\circ$.

Why is it incorrect to say the tangent graph has an amplitude?

Amplitude is defined as the distance from the midline to a maximum or minimum. Since the tangent function goes to infinity and has no maximum or minimum, the concept of amplitude does not apply.

What mistake is made if a student draws $y = \cos(x)$ starting at $(0,0)$?

The student is confusing the cosine graph with the sine graph. Cosine must start at its peak (or trough if reflected) at $x=0$ because $\cos(0) = 1$.

Define the 'Period' of a trigonometric function.

The period is the horizontal length of one complete cycle of the graph. After this distance, the function begins to repeat its pattern of y-values.

What is a vertical asymptote in the context of a tangent graph?

It is a vertical line that the graph approaches but never touches. It occurs at x-values where the function is undefined, specifically where $\cos(x) = 0$.

What does the term 'Midline' refer to in a wave function?

The midline is the horizontal line $y = d$ that represents the average value of the function, situated exactly halfway between the maximum and minimum peaks.

Why do sine and cosine graphs have a range of $[-1, 1]$?

On the unit circle, sine and cosine represent the y and x coordinates of a point. Since the circle has a radius of $1$, these coordinates can never exceed $1$ or be less than $-1$.

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Trigonometric Graphs

Summary

Trigonometric graphs are periodic functions that represent the relationship between angles and the ratios of sides in a right-angled triangle or coordinates on a unit circle. These functions—sine, cosine, and tangent—exhibit repeating patterns (cycles) that are fundamental in modeling oscillating phenomena such as sound waves, light, and tidal movements.

1. Definition & Core Concepts

Periodic Functions: Trigonometric graphs are characterized by their repetitive nature, where the function values repeat at regular intervals known as the period.
Amplitude: This represents the maximum displacement from the horizontal midline to a peak (or trough). For the basic functions $y = \sin(x)$ and $y = \cos(x)$ , the amplitude is $1$ .
Midline (Principal Axis): The horizontal line that runs through the center of the graph, halfway between the maximum and minimum values.
Frequency: The number of cycles the function completes within a specific interval (usually $360^\circ$ or $2\pi$ radians).

A diagram of a sine wave illustrating the concepts of amplitude, period, and the midline.

2. Underlying Principles

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions