Graphs of Trigonometric Functions

Periodic Functions: Trigonometric functions are periodic, meaning they repeat their values in regular intervals called the period. For $y = \sin(x)$ and $y = \cos(x)$, the standard period is $360^\circ$ (or $2\pi$ radians), while for $y = \tan(x)$, it is $180^\circ$ (or $\pi$ radians).

Amplitude and Range: The amplitude is the maximum displacement from the center line; for basic sine and cosine graphs, this is $1$. The range of $\sin(x)$ and $\cos(x)$ is $[-1, 1]$, whereas $\tan(x)$ has a range of $(-\infty, \infty)$.

Asymptotes: The tangent function is undefined at specific points where the cosine value is zero (e.g., $90^\circ, 270^\circ$). At these points, the graph features vertical asymptotes, which the curve approaches but never touches.

Unit Circle Connection: The graphs are derived from the unit circle, where the y-coordinate of a point on the circle represents $\sin(\theta)$ and the x-coordinate represents $\cos(\theta)$. This explains why the values oscillate between $1$ and $-1$.

Symmetry Properties: The sine function is an odd function, meaning $\sin(-x) = -\sin(x)$, which results in rotational symmetry about the origin. The cosine function is an even function, meaning $\cos(-x) = \cos(x)$, resulting in reflectional symmetry across the y-axis.

Tangent Periodicity: Unlike sine and cosine, the tangent function repeats every $180^\circ$. This is because $\tan(x) = \frac{\sin(x)}{\cos(x)}$, and the ratio of sine to cosine repeats its sign and magnitude pattern twice as fast as the individual functions.

Vertical Stretch ($y = a f(x)$): Multiplying the function by a constant $a$ stretches the graph vertically by a scale factor of $a$. This changes the amplitude to $|a|$ and the range to $[-a, a]$.

Horizontal Stretch ($y = f(bx)$): Multiplying the input variable $x$ by a constant $b$ results in a horizontal stretch or squash by a scale factor of $\frac{1}{b}$. The new period is calculated as $\text{New Period} = \frac{\text{Original Period}}{b}$.

Phase Shift ($y = f(x + c)$): Adding a constant $c$ inside the function shifts the graph horizontally. A positive $c$ shifts the graph to the left, while a negative $c$ shifts it to the right.

Vertical Translation ($y = f(x) + d$): Adding a constant $d$ outside the function shifts the entire graph up or down. This changes the principal axis (the center line) from $y = 0$ to $y = d$.

Phase Relationship: The cosine graph is simply a sine graph shifted $90^\circ$ to the left. Mathematically, $\cos(x) = \sin(x + 90^\circ)$.

Asymptote Locations: Tangent asymptotes occur at $x = 90^\circ + 180^\circ k$ (where $k$ is an integer), which are the points where $\cos(x) = 0$.

Finding Multiple Solutions: When solving $\sin(x) = k$, use the graph's symmetry to find the second solution within the first period. For sine, if $x_1$ is a solution, $180^\circ - x_1$ is also a solution. For cosine, if $x_1$ is a solution, $360^\circ - x_1$ is another.

Sketching Accuracy: Always mark five key points for one period of sine or cosine: the start, the first intercept, the peak, the second intercept, and the trough. For tangent, always draw the asymptotes as dashed lines first.

Check the Interval: Exams often specify a domain (e.g., $-180^\circ \le x \le 180^\circ$). Ensure your sketch and solutions are strictly within this range, as periodic functions have infinite solutions otherwise.

Scale Factor Awareness: A common error is applying a horizontal stretch factor of $n$ as a multiplication of the x-coordinates. Remember that $y = \sin(nx)$ means you multiply x-coordinates by $\frac{1}{n}$ (squashing the graph if $n > 1$).