Trigonometric Graphs and Periodic Functions

The Fundamental Graphs of Sine and Cosine

The sine and cosine functions are periodic, meaning they repeat their values in regular intervals. For $y = \sin x$ and $y = \cos x$, the period is $360^{\circ}$ or $2\pi$ radians. When sketching these functions, it is vital to recognise their specific symmetries. The sine function is an odd function, meaning it has rotational symmetry about the origin: $\sin(-x) = -\sin x$. The cosine function is an even function, meaning it has reflectional symmetry in the $y$ axis: $\cos(-x) = \cos x$.

When sketching over large ranges, such as $-720^{\circ} < x \leq 720^{\circ}$, you will see four complete cycles of the wave. Over the range $-2\pi < x \leq 2\pi$, you will see two complete cycles. On the same axes, $y = \sin x$ and $y = \cos x$ intersect where their values are equal. In the range $-2\pi < x \leq 2\pi$, $\cos x = \sin x$ occurs at $x = \pi/4$ and $x = -3\pi/4$. These correspond to where $\tan x = 1$. Conversely, $\cos x = -\sin x$ occurs where $\tan x = -1$, which are the values $x = 3\pi/4$ and $x = -\pi/4$.

Graph Transformations: Amplitude and Period

Modifying the basic trigonometric functions changes their physical appearance on a graph.

1. Vertical Stretching: In the function $y = 2 \sin x$, the number 2 is the amplitude. The graph is stretched vertically by a factor of 2, so the peaks reach 2 and the troughs reach -2.

2. Horizontal Compression: In the function $y = \sin 2x$, the number 2 affects the frequency. The graph is compressed horizontally by a factor of 2, meaning the period is halved from $360^{\circ}$ to $180^{\circ}$ (or $\pi$).

Phase Shifts and Compound Transformations

When a constant is added inside the argument of the function, it results in a horizontal shift (a phase shift). However, the way the constant is written significantly changes the result. Consider these four examples over the range $-2\pi < x \leq 2\pi$:

• $y = \sin(2x + \pi/6)$: Here, the period is $\pi$ and the graph is shifted to the left. The shift is not $\pi/6$. To find the shift, we set the argument to zero: $2x + \pi/6 = 0$, which gives $x = -\pi/12$.
• $y = \sin(2x - \pi/6)$: This is shifted to the right by $\pi/12$.
• $y = \sin 2(x + \pi/6)$: Because the 2 is factorised out, the horizontal shift is exactly $\pi/6$ to the left.
• $y = \sin 2(x - \pi/6)$: This is shifted exactly $\pi/6$ to the right.

Trigonometric Functions as Projection Operators

A powerful way to understand sine and cosine is to view them as projection operators. They project a line of positive length $b$ at an angle $\theta$ onto the axes. The cosine function projects the line onto the $x$ axis, while the sine function projects the line onto the $y$ axis.

In the diagram above, the horizontal projection is $b \cos \theta$. Because $\theta$ is an acute angle in the first quadrant, $b \cos \theta$ is positive.

In the second diagram, the angle $\theta$ is obtuse (between $90^{\circ}$ and $180^{\circ}$). Here, the projection $b \cos \theta$ onto the $x$ axis points in the negative direction, so the value is negative.

For sine, the projection is onto the $y$ axis. In the third diagram, where the angle is in the first or second quadrant, the vertical projection $b \sin \theta$ is positive.

In the fourth diagram, if the angle $\theta$ is between $180^{\circ}$ and $360^{\circ}$, the vertical projection $b \sin \theta$ points downwards, making the value negative. This logic explains the sign of trigonometric functions across the four quadrants of a CAST diagram.

The Tangent Function and Projections

The tangent function converts a projection on the $x$ axis into a projection on the $y$ axis. We can express this relationship as $y = x \tan \theta$.

In the first diagram, both $x$ and $y$ are positive, so $\tan \theta$ is positive. In the second diagram, if $x$ is negative and $y$ is positive (second quadrant), the relationship $y = x \tan \theta$ implies that $\tan \theta$ must be negative. Between $90^{\circ}$ and $180^{\circ}$, $\tan \theta$ is indeed negative. Similarly, between $270^{\circ}$ and $360^{\circ}$ (fourth quadrant), $x$ is positive and $y$ is negative, so $\tan \theta$ remains negative. This demonstrates how tangent represents the gradient of the line at angle $\theta$.