OBJECTIVES

When you finish this section, you should be able to:

1. Work with properties of trigonometric functions (p. 49)
2. Graph the trigonometric functions (p. 50)

NEED TO REVIEW?

Trigonometric functions are discussed in Appendix A.4, pp. A-27 to A-35.

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A function \(f\) is called periodic if there is a positive number \(p\) with the property that whenever \(x\) is in the domain of \(f\), so is \(x+p\), and \[\bbox[5px, border:1px solid black, #F9F7ED]{\bbox[#FAF8ED,5pt]{f(x + p) = f(x)}} \]

If there is a smallest number \(p\) with this property, it is called the ( fundamental) period of \(f\).

THEOREM Period of Trigonometric Functions

\[\bbox[5px, border:1px solid black, #F9F7ED]{\bbox[#FAF8ED,5pt]{\bbox[5px, border:1px solid black, #F9F7ED]{\begin{array}{lll} {\rm{sin}}(x + 2\pi ) = \sin x& \cos (x + 2\pi ) = \cos x& \tan (x + \pi ) = \tan x \\ \csc (x + 2\pi ) = \csc x& \sec (x + 2\pi ) = \sec x& \cot (x + \pi ) = \cot x \end{array}}}} \]

THEOREM Even-Odd Properties of the Trigonometric Functions

The sine, tangent, cosecant, and cotangent functions are odd, so their graphs are symmetric with respect to the origin.

The cosine and secant functions are even, so their graphs are symmetric with respect to the \(y\)-axis.

NEED TO REVIEW?

The values of the trigonometric functions for select numbers are discussed in Appendix A.4, pp. A-29 and A-31.

Properties of the Sine Function \(f( x) =\sin x\)

• The domain of \(f\) is the set of all real numbers.
• The range of \(f\) consists of all real numbers in the closed interval \(\left[-1,1\right]\).
• The sine function is an odd function, so its graph is symmetric with respect to the origin.
• The sine function has a period of \(2\pi\).
• The \(x\)-intercepts of \(f\) are \(\ldots\), \(-2\pi\), \(-\pi\), \(0\), \(\pi\), \(2\pi\), \(3\pi, \ldots;\) the \(y\)-intercept is \(0\).
• The maximum value of \(f\) is 1 and occurs at \(x=\ldots,-\dfrac{3\pi }{2}, \dfrac{\pi }{2}\), \(\dfrac{5\pi }{2}\), \(\dfrac{9\pi }{2}, \ldots\); the minimum value of \(f\) is \(-1\) and occurs at \(x=\ldots, -\dfrac{\pi}{2},\dfrac{3\pi }{2}\) , \(\dfrac{7\pi }{2}\), \(\dfrac{11\pi }{2}, \ldots\).

Properties of the Cosine Function \(f( x) =\cos x\)

• The domain of \(f\) is the set of all real numbers.
• The range of \(f\) consists of all real numbers in the closed interval \([ -1,1]\).
• The cosine function is an even function, so its graph is symmetric with respect to the \(y\)-axis.
• The cosine function has a period of \(2\pi\).
• The \(x\)-intercepts of \(f\) are \(\ldots, -\dfrac{3\pi }{2}\), \(-\dfrac{\pi }{2}\), \(\dfrac{\pi }{2}\), \(\dfrac{3\pi }{2}\), \(\dfrac{5\pi }{2}, \ldots\); the \(y\)-intercept is \(1\).
• The maximum value of \(f\) is \(1\) and occurs at \(x=\,\ldots ,-2\pi\), \(0\), \( 2\pi\), \(4\pi\), \(6\pi , \ldots\); the minimum value of \(f\) is \(-1\) and occurs at \(x= \ldots,-\pi\), \(\pi\), \(3\pi\), \(5\pi , \ldots\).

Graphing Variations of \({f( x) =\sin x}\) Using Transformations

Use the graph of \(f( x) =\sin x\) to graph \(g( x) =2~\sin x\).

Solution Notice that \(g(x) =2f( x)\), so the graph of \(g\) is a vertical stretch of the graph of \(f( x) =\sin x\). Figure 72 illustrates the transformation.

Figure 72

Graphing Variations of \({f( x) =\cos x}\) Using Transformations

Use the graph of \(f( x) =\cos x\) to graph \(g( x) =\cos ( 3x)\).

Solution The graph of \(g( x) =\cos ( 3x)\) is a horizontal compression of the graph of \(f( x) =\cos x\). Figure 73 shows the transformation.

Figure 73

NOW WORK

Problems 27 and 29.

NEED TO REVIEW?

Trigonometric identities are discussed in Appendix A.4, pp. A-32 to A-35.

Proof

To prove this identity, we use the difference formula for \(\cos (\alpha -\beta )\) with \(\alpha =x\) and \(\beta =\dfrac{\pi }{2}\). \[ \cos \, \left( x-\dfrac{\pi }{2}\!\right) =\cos \, x \,\cos \dfrac{\pi }{2}+\sin \, x \, \sin \dfrac{\pi }{2}=\cos \, x\cdot 0+\sin \, x\cdot 1=\sin \, x \]

Because of this relationship, the graphs of \(y=A\sin ( \omega x)\) or \(y=A\cos ( \omega x)\) are referred to as sinusoidal graphs, and the functions and their variations are called sinusoidal functions.

THEOREM

For the graphs of \(y=A\sin ( \omega x)\) and \(y=A\cos ( \omega x)\), where \(A\neq 0\) and \(\omega >0\), \[\bbox[5px, border:1px solid black, #F9F7ED]{\bbox[#FAF8ED,5pt]{ \hbox{Amplitude} =\vert A\vert \qquad \hbox{Period} =T=\dfrac{2\pi }{\omega }}} \]

Finding the Amplitude and Period of a Sinusoidal Function

Determine the amplitude and period of \(y=-3\sin ( 4\pi x)\).

Solution Comparing \(y=-3\sin (4\pi x)\) to \(y = A\sin ( \omega x)\), we find that \(A = -3\) and \(\omega = 4\pi\). Then, \[ \hbox{Amplitude} =\vert A\vert =\vert {-}3\vert =3 \qquad \hbox{Period} =T=\dfrac{2\pi }{\omega }=\dfrac{2\pi }{4\pi }=\dfrac{1}{2} \]

NOW WORK

Problem 33.

Properties of the Tangent Function \(f( x) =\tan x\)

• The domain of \(f\) is the set of all real numbers, except odd multiples of \(\dfrac{\pi }{2}\).
• The range of \(f\) consists of all real numbers.
• The tangent function is an odd function, so its graph is symmetric with respect to the origin.
• The tangent function is periodic with period \(\pi\).
• The \(x\)-intercepts of \(f\) are \(\ldots, -2\pi\), \(-\pi\), \(0\), \(\pi\), \( 2\pi\), \(3\pi, \ldots\); the \(y\)-intercept is \(0\).

Graphing Variations of \({f( x) =\tan x}\) Using Transformations

Use the graph of \(f(x) = \tan x\) to graph \(g(x) =-\tan\! \left(\! x+\dfrac{\pi }{4}\right)\).

Solution Figure 78 illustrates the steps used in graphing \(g( x) =-\tan\! \left(\! x+\dfrac{\pi }{4} \right)\).

• Begin by graphing \(f( x) =\tan x\). See Figure 78(a).
• Replace the argument \(x\) by \(x+\dfrac{\pi }{4}\) to obtain \(y=\tan\! \left(\! x+\dfrac{\pi }{4} \right)\), which shifts the graph horizontally to the left \(\dfrac{\pi }{4}\) unit, as shown in Figure 78(b).
• Multiply \(\tan\! \left(\! x+\dfrac{\pi }{4} \right)\) by \(-1\), which reflects the graph about the \(x\)-axis, and results in the graph of \(y =-\tan\! \left(\! x+\dfrac{\pi }{4} \right)\), as shown in Figure 78(c).

Figure 78

NOW WORK

Problem 31.