How does the period of the tangent function differ from the sine and cosine functions?

The tangent function has a period of $180^\circ$ (or $\pi$ radians), while sine and cosine both have a period of $360^\circ$ (or $2\pi$ radians). This means the tangent graph repeats its pattern twice as often as the others.

What is the difference between the symmetry of $y = \sin(x)$ and $y = \cos(x)$?

Sine has rotational symmetry about the origin (odd function), meaning $\sin(-x) = -\sin(x)$. Cosine has reflectional symmetry across the y-axis (even function), meaning $\cos(-x) = \cos(x)$.

When comparing the graphs of sine and cosine, what is the horizontal shift between them?

The cosine graph is a horizontal shift (translation) of the sine graph by $90^\circ$ to the left. Specifically, $\cos(x) = \sin(x + 90^\circ)$.

What is a common error when identifying the range of the tangent function?

Students often mistakenly assume tangent is bounded between $-1$ and $1$ like sine and cosine. In reality, tangent has an infinite range $(-\infty, \infty)$ because the ratio can grow indefinitely as the angle approaches an asymptote.

Why is it a mistake to only rely on a calculator when solving $\sin(x) = 0.5$ for $0 \le x \le 360$?

A calculator only provides the 'principal value' ($30^\circ$). Because the sine graph is periodic and symmetrical, there is a second solution ($150^\circ$) that the calculator will not show.

What error occurs if asymptotes are not drawn when sketching $y = \tan(x)$?

Without asymptotes, the graph may incorrectly appear continuous. Asymptotes are essential to show where the function is undefined and how the curve trends toward positive or negative infinity.

Define a 'periodic function' in the context of trigonometry.

A periodic function is a function that repeats its values in regular intervals or periods. For trig functions, this means $f(x + nP) = f(x)$, where $P$ is the period and $n$ is an integer.

What is a vertical asymptote?

A vertical asymptote is a vertical line (like $x = 90^\circ$ for tangent) that the graph approaches but never reaches. It occurs at x-values where the function's output is undefined.

What is the 'principal value' of a trigonometric inverse function?

The principal value is the specific solution returned by a calculator, usually chosen from a standard restricted range (e.g., $-90^\circ$ to $90^\circ$ for sine) to ensure the inverse is a function.

How can the symmetry of the cosine graph be used to find a second solution if the first solution is $\theta$?

Because the cosine graph is symmetrical about the y-axis and repeats every $360^\circ$, a second solution in the first circle can be found using $360^\circ - \theta$.

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Graphs of Trigonometric Functions

Summary

Trigonometric functions are periodic mappings that relate angles to ratios. Their graphs—sine, cosine, and tangent—exhibit repeating patterns (periodicity) and specific symmetries that allow for the identification of multiple solutions within a given interval.

1. Core Properties of Sine and Cosine

Graph showing Sine (solid red) and Cosine (dashed blue) waves. Sine starts at (0,0) while Cosine starts at (0,1).

2. The Tangent Function and Asymptotes

Periodicity: Unlike sine and cosine, $y = \tan(x)$ has a shorter period of $180^\circ$ , repeating its cycle twice as often.
Vertical Asymptotes: The function is undefined at odd multiples of $90^\circ$ (e.g., $\pm 90^\circ, \pm 270^\circ$ ), where the graph approaches infinity but never touches the vertical boundary lines.
Infinite Range: The range of the tangent function is $(-\infty, \infty)$ , meaning it can take any real number value.

3. Symmetry Principles

4. Methodology for Sketching

5. Key Distinctions

6. Exam Strategy & Tips