Trigonometric Functions Graphs and Symmetries for the TMUA

Standard Graphs and Periodicity

For the TMUA, you must be intimately familiar with the standard graphs of the sine, cosine, and tangent functions. This includes knowing their shapes, where they cross the axes, and their repetitive nature. You should be able to sketch these in both degrees and radians.

1. Sine and Cosine: These functions are periodic with a period of $360^{\circ}$ (or $2\pi$ radians). This means the graph repeats its entire cycle every $360^{\circ}$.
2. Tangent: This function is periodic with a shorter period of $180^{\circ}$ (or $\pi$ radians). Notably, $\tan 90^{\circ}$ is not defined, leading to vertical asymptotes at $90^{\circ}$, $270^{\circ}$, and every $180^{\circ}$ thereafter.

Modifying Trigonometric Graphs

Understanding how to modify these functions is a key skill. Consider the following exercises to test your comprehension of how coefficients affect the amplitude and period of the graph. Sketching these for ranges like $-720^{\circ}$ to $720^{\circ}$ or $-2\pi$ to $2\pi$ helps reveal their behavior:

• $y = \sin x$, $y = 2 \sin x$ (vertical stretch), and $y = \sin 2x$ (horizontal squash).
• $y = \cos x$, $y = 2 \cos x$, and $y = \cos 2x$.
• $y = \tan x$, $y = 2 \tan x$, and $y = \tan 2x$.

You should also be able to sketch $y = \sin x$ and $y = \cos x$ on the same axes. By doing so, you can observe where they intersect: for instance, $\sin x = \cos x$ when $x = 45^{\circ}$ or $225^{\circ}$ (within the 0 to 360 degree range), and $\cos x = -\sin x$ in other specific intervals.

Advanced transformations like $y = \sin(2x + \frac{\pi}{6})$ require careful attention to the order of operations. This specific example involves a horizontal squash by a factor of 2 and a translation by $\frac{\pi}{12}$ to the left, which is equivalent to $y = \sin 2(x + \frac{\pi}{12})$.

Sine and Cosine as Projection Operators

A powerful way to interpret trigonometric functions is as projection operators. They project a line of a specific length, originating from the center at an angle $\theta$, onto the $x$ or $y$ axes. In this context, the line always has a positive length.

• The cosine function projects the line onto the $x$-axis.
• The sine function projects the line onto the $y$-axis.

This interpretation helps clarify why these functions change sign in different quadrants, as seen in traditional CAST diagrams.

In the diagram above, the projection onto the $x$-axis ($\cos \theta$) is positive because the line is in the first quadrant.

In the second quadrant, the projection $b \cos \theta$ is negative because it falls on the negative $x$-axis.

Similarly, for sine, the projection onto the $y$-axis is positive when the line points upwards in the first or second quadrants.

In the fourth quadrant, the projection $b \sin \theta$ is negative as it falls below the $x$-axis.

The Tangent Function Interpretation

The tangent function converts $x$-axis projections into $y$-axis projections. Mathematically, it is the ratio of the $y$-projection to the $x$-projection.

Consider the signs in different quadrants. Between $90^{\circ}$ and $180^{\circ}$ (the second quadrant), $\cos \theta$ is negative and $\sin \theta$ is positive, making $\tan \theta$ negative. Between $270^{\circ}$ and $360^{\circ}$ (the fourth quadrant), $\cos \theta$ is positive and $\sin \theta$ is negative, so $\tan \theta$ is again negative.