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Chapter 2: Problem 2

We have seen that \(\cos (-t)=\cos (t)\) and \(\sin (-t)=-\sin (t)\) for every real number \(t\). Now assume that \(t\) is a real number for which \(\tan (t)\) is defined. (a) Use the definition of the tangent function to write a formula for \(\tan (-t)\) in terms of \(\sin (-t)\) and \(\cos (-t)\) (b) Now use the negative arc identities for the cosine and sine functions to help prove that \(\tan (-t)=-\tan (t) .\) This is called the negative arc identity for the tangent function. (c) Use the negative arc identity for the tangent function to explain why the graph of \(y=\tan (t)\) is symmetric about the origin.

Short Answer

Expert verified
(a) The formula for \(\tan(-t)\) in terms of \(\sin(-t)\) and \(\cos(-t)\) is: \( \tan(-t) = \frac{\sin(-t)}{\cos(-t)} \) (b) Using negative arc identities for sine and cosine functions, we prove \(\tan(-t)=-\tan(t)\): \( \tan(-t) = \frac{-\sin(t)}{\cos(t)} = -\tan(t) \) (c) Using the negative arc identity for the tangent function, we conclude that the graph of \(y = \tan(t)\) is symmetric about the origin because: \( f(-t) = -f(t) \)

Step by step solution

01

Expressing \(\tan (-t)\) in terms of \(\sin (-t)\) and \(\cos(-t)\).

We know that the tangent function is the ratio of sine to cosine functions. Therefore, we can write: \( \tan(-t) = \frac{\sin(-t)}{\cos(-t)} \) (b) Using negative arc identities for sine and cosine functions to prove \(\tan (-t) = -\tan(t)\).
02

Find the negative arc identities for sine and cosine functions.

We already know that: \( \sin(-t) = -\sin(t) \) \( \cos(-t) = \cos(t) \)
03

Substitute the negative arc identities in the formula for \(\tan(-t)\).

Now, we can substitute these identities into our expression for \(\tan(-t)\): \( \tan(-t) = \frac{-\sin(t)}{\cos(t)} \)
04

Comparing \(\tan(-t)\) with \(\tan(t)\).

The expression for \(\tan(t)\) is: \( \tan(t) = \frac{\sin(t)}{\cos(t)} \) Comparing the expressions for \(\tan(-t)\) and \(\tan(t)\), we can see that: \( \tan(-t) = -\tan(t) \) This is the negative arc identity for the tangent function. (c) Using the negative arc identity for the tangent function to explain the symmetry of the graph of \(y = \tan(t)\) about the origin.
05

Symmetry about origin.

The graph of a function is symmetric about the origin if: \( f(-x) = -f(x) \)
06

Checking the symmetry of the tangent function using the negative arc identity.

We know that: \( \tan(-t) = -\tan(t) \) This means that for the function \(f(t) = \tan(t)\), it is true that: \( f(-t) = -f(t) \) Thus, the graph of the tangent function is symmetric about the origin.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tangent Function
The tangent function, often written as \( \tan(t) \), is an essential concept in trigonometry. It is defined as the ratio of the sine and cosine functions:
  • \( \tan(t) = \frac{\sin(t)}{\cos(t)} \)
This definition implies that the tangent function is only defined where the cosine function is not equal to zero, as division by zero is undefined.
Understanding this relationship helps us see how the tangent function behaves based on the behavior of the sine and cosine functions.
The negative arc identity for the tangent function also revolves around its definition. By substituting negative values, you see that \( \tan(-t) = -\tan(t) \). This identity highlights the unique properties and symmetries inherent in trigonometric functions, which is particularly useful when analyzing their graphs.
Symmetry of Graphs
One fascinating property of trigonometric functions, particularly the tangent function, is their symmetry.
The graph of the tangent function \( y = \tan(t) \) is symmetric about the origin, meaning it reflects identically across the origin point of the graph.
This occurs because of the negative arc identity:
  • \( \tan(-t) = -\tan(t) \)
For a function to be symmetric about the origin, the condition \( f(-x) = -f(x) \) must hold. Hence, since the nature of the tangent function satisfies this condition across all values where it is defined, its graph exhibits perfect symmetry around the origin.
This symmetry has practical implications in both mathematical theories and real-world applications, aiding visualization and comprehension.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for all values for which the functions are defined. They serve as tools for simplifying expressions and solving trigonometric equations.
The negative angle identities are one subset of these identities, illustrating how functions are affected when angles become negative:
  • \( \cos(-t) = \cos(t) \)
  • \( \sin(-t) = -\sin(t) \)
  • \( \tan(-t) = -\tan(t) \)
These identities show notable properties of trigonometric functions, particularly their periodic nature and symmetry characteristics.
Working with these identities enhances your ability to manipulate trigonometric expressions and solve complex equations more efficiently.
It also provides deeper insight into the geometrical interpretations of the functions as they relate to unit circles and coordinate systems.

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Most popular questions from this chapter

Modeling a Heartbeat. For a given person at rest, suppose the heart pumps blood at a regular rate of about 75 pulses per minute. Also, suppose that the volume of this person's heart is approximately 150 milliliters (ml), and it pushes out about \(54 \%\) its volume with each beat. We will model the volume, \(V(t)\) of blood (in milliliters) in the heart at any time \(t,\) as a sinusoidal function of the form $$ V(t)=A \cos (B t)+D $$ (a) If we choose time 0 to be a time when the heart is full of blood, why is it reasonable to use a cosine function for our model? (b) What is the maximum value of \(V(t) ?\) What is the minimum value of \(V(t) ?\) What does this tell us about the values of \(A\) and \(D ?\) Explain. (c) The frequency of a simple harmonic motion is the number of periods per unit time, or the number of pulses per minute in this example. How is the frequency \(f\) related to the period? What value should \(B\) have? Explain. (d) Draw a graph (without a calculator) of your \(V(t)\) using your values of \(A, B,\) and \(D,\) of two periods beginning at \(t=0\) (e) Clearly identify the maximum and minimum values of \(V(t)\) on the graph. What do these numbers tell us about the heart at these times?

(a) Use the Geogebra Applet with the following web address to explore the relationship between the graph of the cosecant function and the sine function. $$ \text { http: } / / \mathrm{gvsu} . \mathrm{edu} / \mathrm{s} / \mathrm{ObH} $$ In the applet, the graph of \(y=\sin (t)\) is shown and is left fixed. Points on the graph of \(y=\csc (t)\) are generated by using the slider for \(t\). For each value of \(t,\) a vertical line is drawn from the point \((t, \sin (t))\) to the point \((t, \csc (t))\). Notice how these points indicate that the graph of the cosecant function has vertical asymptotes at \(t=0, t=\pi,\) and \(t=2 \pi\). (b) Use a graphing utility to draw the graph of \(y=\csc (x)\) \(-\frac{\pi}{2} \leq\) \(x \leq \frac{\pi}{2}\) and \(-10 \leq y \leq 10 .\) Note: It may be necessary to use \(\csc (x)=\) \(\frac{1}{\sin (x)}\) (c) Use a graphing utility to draw the graph of \(y=\csc (x)\) using \(-\frac{3 \pi}{2} \leq\) \(x \leq \frac{3 \pi}{2}\) and \(-10 \leq y \leq 10\)

This exercise provides a justifiction for the properties of the inverse cosine function on page \(150 .\) Let \(t\) be a real number in the closed interval \([0, \pi]\) and let $$ y=\cos (t) $$ We then see that \(-1 \leq y \leq 1\) and $$ \cos ^{-1}(y)=t $$ (a) Use equations (1) and (2) to rewrite the expression \(\cos ^{-1}(\cos (t))\). (b) Use equations (1) and (2) to rewrite the expression \(\cos \left(\cos ^{-1}(y)\right)\).

If necessary, refer to Section 1.6 to answer the followiing questions. (a) How is the cosecant function defined? (b) What is the domain of the cosecant function? (c) Where will the graph of the cosecant function have vertical asymptotes? (d) What is the period of the cosecant function?

(a) Why does the graph of \(y=\csc (x)\) have vertical asymptotes at \(x=0\), \(x=\pi,\) and \(x=2 \pi\) ? What is the domain of the cosecant function? (b) Why is the graph of \(y=\csc (x)\) above the \(x\) -axis when \(0
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