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Chapter 4: Problem 34

Evaluate each expression without using a calculator, and write your answers in radians. $$\cos ^{-1}\left(-\frac{1}{2}\right)$$

Short Answer

Expert verified
The expression evaluates to \( \frac{2\pi}{3} \) radians.

Step by step solution

01

Identify the Meaning of the Expression

The expression \( \cos^{-1}(-\frac{1}{2}) \) asks for the angle \( \theta \) in the range \([0, \pi]\) such that \( \cos(\theta) = -\frac{1}{2} \). This means we need to find the angle whose cosine is \(-\frac{1}{2}\).
02

Recall the Unit Circle Values

On the unit circle, the cosine of angles in radians corresponding to \(-\frac{1}{2}\) are at \( \frac{2\pi}{3} \) and \( \frac{4\pi}{3} \). However, the range for \( \cos^{-1} \theta \) is \([0, \pi]\) for the principal value, so we need to check which angle fits this range.
03

Choose the Correct Angle

Since \( \frac{2\pi}{3} \) is within the range \([0, \pi]\) and satisfies \( \cos(\frac{2\pi}{3}) = -\frac{1}{2} \), it is the correct angle corresponding to the inverse cosine value. \( \frac{4\pi}{3} \), on the other hand, is outside of this range.

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Key Concepts

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

Understanding the Cosine Function
The cosine function is one of the fundamental trigonometric functions, typically denoted as \( \cos \theta \). It is used to relate the angle in a right-angled triangle to the ratio of the adjacent side over the hypotenuse. In the context of the unit circle, the cosine of an angle \( \theta \) can be interpreted as the x-coordinate of the point on the unit circle at that angle. This means:
  • On a unit circle, every point can be described as \((\cos \theta, \sin \theta)\) where \( \theta \) is the angle from the positive x-axis.
  • The cosine values range between -1 and 1 as they represent the horizontal distance from the origin.
  • At \( \theta = 0 \), \( \cos \theta = 1 \). For \( \theta = \pi/2 \) or 90 degrees, \( \cos \theta = 0 \), and for \( \theta = \pi \), \( \cos \theta = -1 \).
Cosine is symmetrical about the y-axis in the unit circle, highlighting its even function nature, meaning \( \cos(-\theta) = \cos(\theta) \).
The Unit Circle Explained
The unit circle is a vital concept in trigonometry, providing a geometric representation of angles and their corresponding trigonometric values. Defined as a circle with a radius of 1 centered at the origin of a coordinate plane, the unit circle helps in understanding the behavior of trigonometric functions.
  • Each point on the unit circle corresponds to an angle measured in radians from the positive x-axis.
  • It allows us to visualize the values of sine and cosine across different quadrants and angles.
  • Key angles include \( \pi/6, \pi/4, \pi/3, \pi/2, \pi \), and their corresponding coordinates provide the common trigonometric values like 1/2, \( \sqrt{2}/2 \), and \( \sqrt{3}/2 \).
The properties of the unit circle, such as its symmetry and periodicity, are critical in defining inverse trigonometric functions, including \( \cos^{-1} \theta \). These functions return angles for given trigonometric values within specific ranges, making the unit circle a foundational tool in solving trigonometric equations.
Understanding Radian Measure
Radian measure is the standard unit for measuring angles, based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians offer a more natural approach by relating the angle to the arc length of the circle.
  • One complete revolution around a circle is \( 2\pi \) radians, analogous to 360 degrees.
  • A right angle is \( \pi/2 \) radians, equivalent to 90 degrees.
  • Radian measure can be converted to degrees using the relation: \( 180^\circ = \pi \) radians.
This measure is particularly useful in calculus and physics, as it simplifies the derivative and integral calculations involving trigonometric functions. When dealing with inverse trigonometric functions like \( \cos^{-1} \theta \), understanding radian measure aids in picking the correct angle within specified ranges, reinforcing its importance in advanced mathematical contexts.

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Most popular questions from this chapter

Graph each of the following over the given interval. Label the axes so that the amplitude and period are easy to read. $$ y=-2 \cos (-3 x), 0 \leq x \leq 2 \pi $$

The problems that follow review material we covered in Section 3.2. Reviewing these problems will help you with the next section. Evaluate each of the following if \(x\) is \(\pi / 2\) and \(y\) is \(\pi / 6\). $$ \sin \left(x+\frac{\pi}{2}\right) $$

Use a calculator to approximate each value to four decimal places. \(\cos 10^{4}\)

Problems 69 through 76 will help prepare you for the next section. Use your graphing calculator to graph each family of functions for \(-2 \pi \leq x \leq 2 \pi\) together on a single coordinate system. (Make sure your calculator is set to radian mode.) What effect does the value of \(k\) have on the graph? $$ y=k+\sin x \quad \text { for } k=0,2,4 $$

Any object or quantity that is moving with a periodic sinusoidal oscillation is said to exhibit simple harmonic motion. This motion can be modeled by the trigonometric function $$ y=A \sin (\omega t) \quad \text { or } \quad y=A \cos (\omega t) $$ where \(A\) and \(\omega\) are constants. The constant \(\omega\) is called the angular frequency. A mass attached to a spring oscillates upward and downward. The displacement of the mass from its equilibrium position after \(t\) seconds is given by the function \(d=-3.5 \cos (2 \pi t)\), where \(d\) is measured in centimeters (Figure 13). a. Sketch the graph of this function for \(0 \leq t \leq 5\). b. What is the furthest distance of the mass from its equilibrium position? c. How long does it take for the mass to complete one oscillation?
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