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Chapter 5: Problem 9

evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. $$ \cos \pi $$

Short Answer

Expert verified
\(\cos(\pi) = -1\)

Step by step solution

01

Understanding the trigonometric function

The cosine function returns the x-coordinate of the point on the unit circle determined by the angle. In this case, the value of the angle is \(\pi\), which is also known as 180 degrees. The cosine function is positive in the first and fourth quadrants and negative in the second and third quadrants. As such, since the angle \(\pi\) lies in the second quadrant, the cosine of \(\pi\) should be negative.
02

Evaluating \(\cos(\pi)\)

The x-coordinate of the point on the unit circle determined by the angle \(\pi\) is -1. Thus, \(\cos(\pi) = -1\).

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Key Concepts

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

Understanding the Unit Circle
The unit circle is a fundamental concept in trigonometry. It's a circle with a radius of 1, centered at the origin of a coordinate system. This circle is particularly useful because it allows us to define trigonometric functions for all angles using simple geometric relationships. Each point on the unit circle corresponds to some angle \(\theta\) measured from the positive x-axis.
  • The x-coordinate of a point gives you the cosine value of the angle.
  • The y-coordinate provides the sine value of the angle.
The unit circle helps us understand how the trigonometric functions cycle through their values as angles change. Angles can be measured in radians or degrees, and the unit circle accommodates both systems.
Defining Quadrantal Angles
Quadrantal angles are those angles that lie on the x-axis or the y-axis of the coordinate plane. These special angles make key intersecting points with the unit circle.
  • Examples of quadrantal angles are 0, \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\) in radians.
  • In degrees, these correspond to 0掳, 90掳, 180掳, and 270掳 respectively.
Quadrantal angles are significant because their trigonometric values, particularly sine and cosine, are integer values or sometimes zero. This fact simplifies many calculations and helps in evaluating functions at these angles.
Exploring the Cosine Function
The cosine function is one of the primary trigonometric functions, closely related to the unit circle. It measures the horizontal distance from the origin to a point on the circle.
  • The cosine of an angle \(\theta\) is represented as \( ext{cos}(\theta)\).
  • It gives the x-coordinate of the point where the terminal side of the angle intersects the unit circle.
  • Cosine values range from -1 to 1 as you move around the unit circle.
Understanding how cosine values change as we move through different quadrants in the unit circle helps in predicting the sign of cosine. It's positive in the first and fourth quadrants and negative in the second and third quadrants.
Identifying the X-Coordinate
The x-coordinate plays an integral role when evaluating the cosine function on the unit circle. When an angle is given, say \(\pi\), the x-coordinate represents the cosine of that angle.
  • For the angle \(\pi\) (or 180掳), the corresponding point on the unit circle is (-1, 0).
  • Thus, the x-coordinate here is -1.
  • This means \( ext{cos}(\pi) = -1\).
Understanding the link between x-coordinates and cosine values helps you evaluate trigonometric functions efficiently, especially for quadrantal angles like \(\pi\). This approach streamlines solving tasks involving these common angles.

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Most popular questions from this chapter

Your neighborhood movie theater has a 25 -foot-high screen located 8 feet above your eye level. If you sit too close to the screen, your viewing angle is too small, resulting in a distorted picture. By contrast, if you sit too far back, the image is quite small, diminishing the movie's visual impact. If you sit \(x\) feet back from the screen, your viewing angle, \(\theta,\) is given by $$\theta=\tan ^{-1} \frac{33}{x}-\tan ^{-1} \frac{8}{x}$$ (GRAPH CANNOT COPY) Find the viewing angle, in radians, at distances of 5 feet, 10 feet, 15 feet, 20 feet, and 25 feet.

Use a graphing utility to graph two periods of the function. $$y=0.2 \sin \left(\frac{\pi}{10} x+\pi\right)$$

Use a graphing utility to graph two periodsof the function. Use a graphing utility to graph \(y=\sin x\) and \(y=x-\frac{x^{3}}{6}+\frac{x^{5}}{120}\) in a \(\left[-\pi, \pi, \frac{\pi}{2}\right]\) by \([-2,2,1]\) viewing rectangle. How do the graphs compare?

How do we measure the distance between two points, \(A\) and \(B,\) on Earth? We measure along a circle with a center, \(C,\) at the center of Earth. The radius of the circle is equal to the distance from \(\mathrm{C}\) to the surface. Use the fact that Earth is a sphere of radius equal to approximately 4000 miles to solve Exercises 93-96. If \(\theta=10^{\circ},\) find the distance between \(A\) and \(B\) to the nearest mile.

Graph each pair of functions in the same viewing rectangle. Use your knowledge of the domain and range for the inverse trigonometric function to select an appropriate viewing rectangle. How is the graph of the second equation in cach exercise related to the graph of the first equation? $$ y=\sin ^{-1} x \text { and } y=\sin ^{-1}(x+2)+1 $$
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