Problem 25 - Algebra and Trigonometry

Question

9–32 Find the exact value of the trigonometric function. $$\cos \left(-\frac{7 \pi}{3}\right)$$

Step-by-Step Solution

Answer

\( \frac{1}{2} \)

Step 1: Understanding Negative Angles

To find the cosine of a negative angle, recall that cosine is an even function, meaning that \( \cos(-\theta) = \cos(\theta) \). Therefore, \( \cos \left(-\frac{7\pi}{3}\right) = \cos \left(\frac{7\pi}{3}\right) \).

Step 2: Rewriting the Angle

The angle \( \frac{7\pi}{3} \) is larger than \( 2\pi \). We need to find a coterminal angle between \( 0 \) and \( 2\pi \). Subtract \( 2\pi \) from \( \frac{7\pi}{3} \): \( \frac{7\pi}{3} - 2\pi = \frac{7\pi}{3} - \frac{6\pi}{3} = \frac{\pi}{3} \).

Step 3: Evaluating the Cosine Function

Now find \( \cos\left(\frac{\pi}{3}\right) \). From trigonometric tables or the unit circle, \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \).

Key Concepts

Negative Angles

When dealing with trigonometric functions, it's important to understand how they interact with negative angles. Negative angles are simply angles measured in the clockwise direction, as opposed to the standard counter-clockwise direction. In trigonometry, cosine is considered an even function, which means that it has a unique property: the cosine of a negative angle is equal to the cosine of the positive angle. This can be mathematically expressed as:\( \cos(-\theta) = \cos(\theta) \)Knowing this property simplifies problems involving negative angles. For example, finding the cosine of \(-\frac{7\pi}{3}\) is the same as finding \(\cos\left(\frac{7\pi}{3}\right)\). This is because the negative sign does not affect the value of cosine.This concept helps us focus on simply finding the equivalent positive angle, allowing us to utilize other trigonometric tools like the unit circle to solve the problem more efficiently. Once you grasp this concept, handling negative angles becomes much easier.

Unit Circle

The unit circle is a fundamental concept in trigonometry that provides a visual method to understand trigonometric functions and how they work. It is a circle with a radius of 1 centered at the origin of a coordinate plane.A point on the unit circle at an angle \(\theta\) from the positive x-axis has coordinates \((\cos(\theta), \sin(\theta))\). These coordinates represent the cosine and sine of the angle \(\theta\), respectively. For example, on the unit circle, the angle \(\frac{\pi}{3}\) corresponds to the point \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\). Therefore, \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\).Understanding the unit circle allows you to easily identify values for basic angles because these values repeat in a regular pattern as you move around the circle. This is especially handy when dealing with larger angles or when converting negative angles to their positive counterparts. In essence, the unit circle offers a straightforward way to evaluate trigonometric functions like cosine.

Coterminal Angles

Coterminal angles can be a useful tool when trying to find equivalent angles that produce the same cosine value. These angles share the same terminal side when drawn in standard position, effectively giving them identical trigonometric function values. To find a coterminal angle, you can either add or subtract full rotations of \(2\pi\) radians (which is equivalent to 360 degrees) from the given angle. This process finds an equivalent angle within the standard range of \([0, 2\pi)\) for radians or \([0, 360^\circ)\) for degrees.In our problem, \(\frac{7\pi}{3}\) is clearly more than \(2\pi\), which means it lies outside the first full rotation of the unit circle. By subtracting \(2\pi\), we find the coterminal angle\[ \frac{7\pi}{3} - 2\pi = \frac{\pi}{3} \]With this angle, we can use the unit circle to conveniently find \(\cos\left(\frac{\pi}{3}\right)\), as this coterminal angle has the same cosine value as \(\frac{7\pi}{3}\). Coterminal angles let us simplify problems by working with smaller, more familiar angles when evaluating trigonometric functions.