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Chapter 1: Problem 48

Without using a calculator, evaluate, if possible, the following expressions. $$\cos ^{-1}(-1)$$

Short Answer

Expert verified
Answer: \(\pi\)

Step by step solution

01

Recall the inverse cosine function definition

Recall that the inverse cosine function, denoted as \(\cos^{-1}(x)\), is the angle whose cosine is x. In this case, we are looking for an angle (in radians) whose cosine is -1.
02

Use the unit circle to find the angle

On the unit circle, cosine represents the x-coordinate value at a given angle. We want to find the angle where the x-coordinate is -1. We know that cosine function is -1 at one particular point on the unit circle. This point is located at \(\pi\), as cos(蟺) = -1 (or 180掳 in degrees).
03

Write the final answer

Since the x-coordinate is -1 at the angle \(\pi\), we determined that: $$\cos ^{-1}(-1) = \pi$$

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Key Concepts

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

Cosine
Cosine is a fundamental concept in trigonometry that deals with the relationship between the angle and the sides of a right triangle.
  • In a right triangle, cosine is defined as the ratio of the length of the adjacent side to the hypotenuse.
  • Mathematically, it is expressed as: \[\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\]where \(\theta\) is the angle in question.
  • In the context of the unit circle, the cosine value corresponds to the x-coordinate of a point on the circle.
  • Cosine values range between -1 and 1.
When considering inverse trigonometric functions, like \( \cos^{-1}(x) \), the function provides the angle whose cosine is \( x \). For instance, \(\cos^{-1}(-1)\) seeks an angle where the cosine value is -1.Understanding the cosine is crucial because it helps in visualizing geometric relationships in trigonometry and is widely applied in fields such as physics and engineering.
Unit Circle
The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is a key tool in trigonometry to define the trigonometric functions such as sine and cosine.
  • On the unit circle, each angle \(\theta\) corresponds to a point \((x, y)\).
  • The x-coordinate of this point gives the cosine of the angle \(\theta\), and the y-coordinate gives the sine.
  • Therefore, for an angle \(\theta\), the coordinates \((\cos(\theta), \sin(\theta))\) are obtained.
In our problem, we are interested in \(\cos^{-1}(-1)\). On the unit circle, an x-coordinate of -1 occurs at \(\theta = \pi\) (or 180掳). Visualizing the unit circle helps provide intuition about why the cosine of \(\pi\) is -1. This is because at \(\pi\) radians, the point is at the far left of the circle, which corresponds to an x-coordinate of -1.
Radians
Radians are a way of measuring angles based on the radius of a circle. Instead of measuring in degrees, which divide a circle into 360 equal parts, radians focus on the relationship between the radius and the arc length.
  • One full circle encompasses \(2\pi\) radians.
  • Therefore, all 360 degrees translate into \(2\pi\) radians.
  • This implies that \(\pi\) radians corresponds to 180掳.
  • In trigonometry, radians often provide a more natural measure of angles, especially when dealing with periodic functions.
In the execution of our problem, we see that \(\cos^{-1}(-1) = \pi\), which means the angle measures \(\pi\) radians or 180掳, where the cosine reaches the value of -1. Understanding radians helps in analyzing trigonometric functions smoothly across different contexts, especially in integrating and differentiating functions, where angle measures need to naturally relate to the properties of circles.

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Most popular questions from this chapter

A single slice through a sphere of radius \(r\) produces a cap of the sphere. If the thickness of the cap is \(h,\) then its volume is \(V=\frac{1}{3} \pi h^{2}(3 r-h) .\) Graph the volume as a function of \(h\) for a sphere of radius \(1 .\) For what values of \(h\) does this function make sense?

Use analytical methods to find the following points of intersection. Use a graphing utility only to check your work. Find the point(s) of intersection of the parabolas \(y=x^{2}\) and \(y=-x^{2}+8 x\)

Prove that the area of a sector of a circle of radius \(r\) associated with a central angle \(\theta\) (measured in radians) is \(A=\frac{1}{2} r^{2} \theta\)

Make a sketch of the given pairs of functions. Be sure to draw the graphs accurately relative to each other. $$y=x^{4} \text { and } y=x^{6}$$

Consider the quartic polynomial \(y=f(x)=x^{4}-x^{2}\). a. Graph \(f\) and estimate the largest intervals on which it is one-to-one. The goal is to find the inverse function on each of these intervals. b. Make the substitution \(u=x^{2}\) to solve the equation \(y=f(x)\) for \(x\) in terms of \(y .\) Be sure you have included all possible solutions. c. Write each inverse function in the form \(y=f^{-1}(x)\) for each of the intervals found in part (a).
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