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4.3 Graphs of the Trigonometric Functions

A table of values

From the <, we can construct a table of values of the trigonometric functions for the multiples of \(45^\circ\) or , in radians, \(\frac{\pi}{4}\), between \(0\) and \(2\pi\) inclusive. We will generally use radians instead of degrees since radians are used in calculus.

\(x\) \(0\) \( \frac{\pi}{4} \) \( \frac{\pi}{2} \) \( \frac{3\pi}{4} \) \( \pi \) \( \frac{5\pi}{4} \) \( \frac{3\pi}{2} \) \( \frac{7\pi}{4} \) \( 2\pi \)
\(\sin x\) \( 0 \) \( \frac{\sqrt{2}}{2} \) \( 1 \) \( \frac{\sqrt{2}}{2} \) \( 0 \) \( -\frac{\sqrt{2}}{2} \) \( -1 \) \( -\frac{\sqrt{2}}{2} \) \( 0 \)
\(\cos x\) \( 1 \) \( \frac{\sqrt{2}}{2} \) \( 0 \) \( -\frac{\sqrt{2}}{2} \) \( -1 \) \( -\frac{\sqrt{2}}{2} \) \( 0 \) \( \frac{\sqrt{2}}{2} \) \( 1 \)
\(\tan x\) \( 0 \) \( 1 \) \( \cdots \) \( -1 \) \( 0 \) \( 1 \) \( \cdots \) \( -1 \) \( 0 \)
\(\csc x\) \( \cdots \) \( \sqrt{2} \) \( 1 \) \( \sqrt{2} \) \( \cdots \) \( -\sqrt{2} \) \( -1 \) \( -\sqrt{2} \) \( \cdots \)
\(\sec x\) \( 1 \) \( \sqrt{2} \) \( \cdots \) \( -\sqrt{2} \) \( -1 \) \( -\sqrt{2} \) \( \cdots \) \( \sqrt{2} \) \( 1 \)
\(\cot x\) \( \cdots \) \( 1 \) \( 0 \) \( -1 \) \( \cdots \) \( 1 \) \( 0 \) \( -1 \) \( \cdots \)

Some of the trigonometric functions are undefined for certain angles. This is indicated in the table with three dots (\(\cdots)\). The graphs of those functions have transitive vertical asymptotes at those values. Remember that a vertical asymptote is transitive if the graph approaches the top half of the vertical asymptote on one side and the bottom half on the other side.

Exercise 4.3.1

Using \(\frac{\sqrt{2}}{2}\approx0.7\) sketch the graph of \(y=\sin x\) between \(x=0\) and \(x=2\pi\).

See Solution

Exercise 4.3.2

Sketch the graph of \(y = \cos x\) between \(x = 0\) and \(x = 2\pi\)

See Solution

Exercise 4.3.3

Construct a table of values of \(\sin x\) between \(x = 2\pi\) and \(x = 4\) in increments of \(\frac{\pi}{4}\).

Sketch the graph of \(y = \sin x\) between \(x = 0\) and \(x = 4\pi\).

See Solution

Trigonometric functions are periodic

Notice that the sine graph between \(x = 2\pi\) and \(x = 4\pi\) repeats the graph between \(x = 0\) and \(x = 2\pi\).

The graph of the sine function consists of infinite repetitions of the portion of the graph between \(x = 0\) and \(x = 2\pi\). The sine graph, as well as the other trigonometric functions, satisfies the property that \(f ( x + 2\pi) = f ( x )\) for all \(x\) in the domain of the function. This is because the angle \(x\) and the angle \(x + 2\pi\) in standard position have the same terminal side. Functions with the property that, for some constant \(d\), \(f ( x + d ) = f ( x )\) for all \(x\) in the domain of the function are called periodic functions. For a periodic function \(f\), the smallest value \(d\) for which \(f ( x + d ) = f ( x )\) for all \(x\) in the domain of \(f\) is called the period of \(f\).

The sine and cosine functions are periodic functions with period \(2\pi\).

Notice that the graph of cosine looks like the graph of sine shifted \(\frac{\pi}{2}\) units to the left. In other words, \(\cos x = \sin ( x + \frac{\pi}{2} )\).

Now we will investigate the graphs of the reciprocal functions of sine and cosine—the graphs of cosecant and secant.

\(\csc x = 1 / \sin x\) and \(\sec x = 1 / \cos x\) .

Whenever \(\sin x = 1\), then \(\csc x = 1\). Whenever \(\cos x = 1\), then \(\sec x = 1\).

Whenever \(\sin x = -1\), then \(\csc x = -1\). Whenever \(\cos x = -1\), then \(\sec x = -1\).

Whenever \(\sin x = 0\), \(\csc x\) is undefined, and the graph of cosecant has a transitive vertical asymptote.

Whenever \(\cos x = 0\), \(\sec x\) is undefined, and the graph of secant has a transitive vertical asymptote.

Since the sine and cosine functions have period \(2\pi\), then so do their reciprocal functions cosecant and secant.

Exercise 4.3.4

Using \(\sqrt{2}\approx 1.4\), sketch one cycle of the graph of \(y = \csc x\) between \(x = 0\) and \(x = 2\pi\). Sketch the vertical asymptotes as dotted lines. It may be useful to also sketch the graph of \(y=\sin x\) as a dotted line to see the reciprocal relationship between the two graphs.

See Solution

Exercise 4.3.5

Sketch the graph of \(y = \sec x\) between \(x = 0\) and \(x = 2\pi\). Sketch the vertical asymptotes and the graph of \(y=\cos x\) as dotted lines.

See Solution

The graph of tangent

Next, we will consider the graph of tangent function.

An important fact about the tangent of an angle is that it equals the slope of the terminal side of the angle.

Suppose \(x\) is an angle whose terminal side is not a vertical line (else it will not have a slope and its tangent will be undefined). If we add \(\pi\) radians to the angle \(x\) we will have an angle \(x+\pi\) whose terminal side lies on the opposite side of the origin from the terminal side of angle \(x\) along the same line through the origin! Since the tangent of an angle equals the slope of the terminal side of the angle (in standard position) this means that \(\tan(x+\pi)=\tan x\). So the period of the tangent (as well as its reciprocal the cotangent) is \(\pi\) unlike the other four trigonometric functions whose periods are \(2\pi\).

Recall that \(\tan x = \sin x / \cos x\). The tangent graph has a transitive vertical asymptote for any value of \(x\) for which \(\cos x=0\).

Exercise 4.3.6

Sketch two cycles of the graph of \(y = \tan x\) between \(x = 0\) and \(x = 2\pi\). Sketch the vertical asymptotes as dotted lines.

See Solution

The graph of cotangent

The cotangent is the reciprocal of the tangent function: \(\cot x = 1 / \tan x\)

Whenever \(\tan x = 1\), \(\cot x = 1\). Whenever \(\tan x = -1\), \(\cot x = -1\).

Whenever \(\tan x = 0\), cotangent is undefined and its graph has a transitive vertical asymptote.

Whenever tangent is undefined, cotangent equals zero.

Exercise 4.3.7

Sketch two cycles of the graph of \(y = \cot x\) between \(x = 0\) and \(x = 2\). Sketch the vertical asymptotes as dotted lines.

See Solution