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Chapter 8: Problem 41

Find the value. $$\cos \left(-\frac{2 \pi}{3}\right)$$

Short Answer

Expert verified
- \(\frac{1}{2}\)

Step by step solution

01

Identify the angle in standard position

The angle \(- \frac{2 \pi}{3}\) is negative. To convert it to its equivalent positive angle, add \(2\pi \) to \(- \frac{2 \pi}{3}\). \(- \frac{2 \pi}{3} + 2 \pi = \frac{4 \pi}{3}\).
02

Understand the reference angle

The angle \(\frac{4 \pi}{3}\) is in the third quadrant. The reference angle is \(\frac{4 \pi}{3} - \pi = \frac{\pi}{3}\).
03

Determine the cosine value

In the third quadrant, the cosine function is negative. The cosine of the reference angle is \(\frac{\pi}{3}\), which is \(\frac{1}{2}\). Therefore, \(\text{cos} \left(\frac{4 \pi}{3}\right) = - \frac{1}{2}\).

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Key Concepts

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

trigonometric functions
Trigonometric functions are crucial in understanding relationships in triangles and circles. One of these functions is the cosine function, which is often written as \(\text{cos}(胃)\). This function relates the angle in a right triangle to the ratio of the adjacent side to the hypotenuse. In a unit circle, cosine represents the x-coordinate of a point on the circle corresponding to an angle measured from the positive x-axis. Common trigonometric functions include sine (\(\text{sin}(胃)\)), tangent (\(\text{tan}(胃)\)), and of course, cosine. Becoming familiar with these functions and their properties is essential for mastering concepts in trigonometry.
reference angle
A reference angle is the smallest angle between the terminal side of the given angle and the x-axis. Reference angles help in simplifying the calculation of trigonometric functions for angles that are not in the first quadrant. To find a reference angle:
  • If the angle is in the second quadrant, subtract it from \(蟺\).
  • If it's in the third quadrant, subtract \(蟺\) from the angle.
  • For the fourth quadrant, subtract the angle from \(2蟺\).
For example, in our exercise, the angle \( \frac{4 \pi}{3} \) lies in the third quadrant. The reference angle is found by subtracting \( \pi \) from \( \frac{4 \pi}{3} \), giving us \( \frac{\pi}{3} \). This reference angle can then be used to find the cosine value.
cosine in different quadrants
The sign of the cosine function changes depending on the quadrant in which the angle lies:
  • In the first quadrant, cosine is positive.
  • In the second quadrant, cosine is negative.
  • In the third quadrant, cosine is also negative.
  • In the fourth quadrant, cosine is positive again.
Understanding this helps in evaluating trigonometric functions without confusion. For instance, the angle \( \frac{4 \pi}{3} \) is in the third quadrant, where the cosine function is negative. Given that the reference angle is \( \frac{\pi}{3} \) with a standard cosine value of \( \frac{1}{2} \), the cosine of \( \frac{4 \pi}{3} \) is determined to be \( - \frac{1}{2} \), reflecting the correct sign for the third quadrant.

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Most popular questions from this chapter

Raise the number to the given power and write standard notation for the answer. $$\left[2\left(\cos 10^{\circ}+i \sin 10^{\circ}\right)\right]^{9}$$

Appeared first in Exercise Set \(8.5,\) where we used the law of cosines and the law of sines to solve the applied problems. For this exercise set, solve the problem using the vector form $$\mathbf{v}=|\mathbf{v}|[(\cos \theta) \mathbf{i}+(\sin \theta) \mathbf{j}]$$ An airplane has an airspeed of \(150 \mathrm{km} / \mathrm{h}\). It is to make a flight in a direction of \(70^{\circ}\) while there is a \(25-\mathrm{km} / \mathrm{h}\) wind from \(340^{\circ} .\) What will the airplane's actual heading be?

Convert to degree measure. $$3 \pi$$

Find the square roots of the number. $$2 \sqrt{2}-2 \sqrt{2} i$$

Find the function value using coordinates of points on the unit circle. $$\sin \frac{5 \pi}{6}$$
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