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Chapter 1: Problem 30

Use reference angles to find the exact value of each expression. $$ \cos (-5 \pi / 3) $$

Short Answer

Expert verified
The exact value is \( \cos \left( -\frac{5 \pi}{3} \right) = \frac{1}{2} \).

Step by step solution

01

- Identify the given angle

The given angle is \( - \frac{5 \pi}{3} \). This angle is negative, indicating that it is measured in the clockwise direction from the positive x-axis.
02

- Convert the negative angle to a positive angle

To find the equivalent positive angle, add \( 2 \pi \) to the given angle: \( - \frac{5 \pi}{3} + 2 \pi \). This simplifies to \( - \frac{5 \pi}{3} + \frac{6 \pi}{3} = \frac{ \pi}{3} \).
03

- Determine the reference angle

The angle \( \frac{ \pi}{3} \) is already in the first quadrant, and it is its own reference angle.
04

- Find the cosine of the reference angle

The cosine of \( \frac{ \pi}{3} \) is known to be \( \frac{1}{2} \).
05

- Conclude with the exact value

Since the reference angle \( \frac{ \pi}{3} \) is in the first quadrant, the value remains positive. Therefore, \( \cos \left( -\frac{5 \pi}{3} \right) = \frac{1}{2} \).

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Key Concepts

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

negative angles
When dealing with trigonometric functions, it's essential to understand the concept of negative angles. A negative angle means measuring the angle in the clockwise direction from the positive x-axis. Imagine a clock: moving the hands in the direction that time goes backward represents a negative angle. For example, \(-\frac{5 \pi}{3}\) means starting from the positive x-axis and rotating clockwise.
angle conversion
Converting negative angles to positive angles is a common step in solving trigonometric problems. To convert a negative angle to its positive equivalent, add \(2 \pi\), since the trigonometric circle repeats every \(2 \pi\). For the angle \(-\frac{5 \pi}{3}\), adding \(2 \pi\) gives:
\(-\frac{5 \pi}{3} + 2 \pi = -\frac{5 \pi}{3} + \frac{6 \pi}{3} = \frac{\pi}{3}\). This makes \(\frac{\pi}{3}\) the positive equivalent of \(-\frac{5 \pi}{3}\).
cosine function
The cosine function (cos) is one of the primary trigonometric functions. It relates the angle of a right triangle to the ratio of the adjacent side over the hypotenuse. When considering the unit circle, the cosine of an angle is the x-coordinate of the point where the terminal side of the angle intersects the unit circle. For example, \( \cos \left( \frac{\pi}{3} \right)\) is the x-coordinate of the point on the unit circle at an angle of \(\frac{\pi}{3}\), which is \(\frac{1}{2}\).
reference angles
Reference angles help simplify the process of finding trigonometric function values for any given angle. A reference angle is always the acute angle formed by the terminal side of the given angle and the x-axis. For instance, in the problem above, \(\frac{\pi}{3}\) is already in the first quadrant and is its own reference angle. Thus, knowing the cosine of the reference angle \(\frac{\pi}{3}\), which is \(\frac{1}{2}\), allows us to find the cosine for the original angle \(-\frac{5 \pi}{3}\). The reference angle directly provides the cosine value, simplifying calculations.

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Most popular questions from this chapter

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