Problem 22 - Precalculus

Question

Use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions. $$\cos \left(-\frac{7 \pi}{4}\right)$$

Step-by-Step Solution

Answer

\(\cos\left(-\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}\)

Step 1: Identify the Angle's Coterminal Angle

To simplify the analysis, convert the angle \(-\frac{7\pi}{4}\) to a positive angle by adding \(2\pi\) repeatedly until we get a positive angle less than \(2\pi\). Knowing a full rotation is \(2\pi\), we add \(2\pi\) to \(-\frac{7\pi}{4}\) to get the coterminal angle:\[-\frac{7\pi}{4} + 2\pi = -\frac{7\pi}{4} + \frac{8\pi}{4} = \frac{\pi}{4}\]The angle \(\frac{\pi}{4}\) is coterminal, so it’s equivalent on the unit circle.

Step 2: Use the Even Property of Cosine

Cosine is an even function, which means \(\cos(-x) = \cos(x)\). Therefore, the cosine of the original angle \(-\frac{7\pi}{4}\) is equal to the cosine of its positive coterminal angle \(\frac{\pi}{4}\):\[\cos\left(-\frac{7\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right)\]

Step 3: Evaluate Cosine at Positive Angle

Now, use the unit circle or known trigonometric values to find \(\cos\left(\frac{\pi}{4}\right)\). The value of cosine at \(\frac{\pi}{4}\) is a standard trigonometric value:\[\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\]

Step 4: Conclusion

From the previous steps, we have established:\[\cos\left(-\frac{7\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\]

Key Concepts

Cosine

Cosine is one of the primary trigonometric functions often used in trigonometry. It arises in various applications, including geometry, physics, and engineering. Cosine corresponds to the x-coordinate of a point on the unit circle.
The unit circle is a circle with a radius of one, centered at the origin of a coordinate plane. The angle, usually denoted in radians, determines a point on this circle, and the cosine of that angle is the x-value of this point.

Key characteristics of cosine include:

• Cosine of 0 is 1 because the angle is along the positive x-axis.
• Cosine decreases as angles move towards \(\pi/2\), where the cosine is 0 since it is along the y-axis.
• The function is periodic, with a period of \(2\pi\); meaning it repeats every \(2\pi\) radians, the circumference of the unit circle.

For the angle \(\frac{\pi}{4}\), \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\.
This value is derived from the 45-45-90 right triangle, where the legs are equal, leading to an equal x and y distribution.

Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides when drawn in standard position. You can find coterminal angles by adding or subtracting full circle rotations to an angle. Full rotation is equivalent to \(2\pi\) radians or 360 degrees.

To determine a coterminal angle, you can apply the formula:

• Given an angle \(\theta\), coterminal angles can be found using \(\theta + 2k\pi\) where \(k\) is an integer.

In the exercise, we start with the angle of \(-\frac{7\pi}{4}\). By adding \(2\pi\), we make it positive to the equivalent \(\frac{\pi}{4}\).
These angles provide flexibility when evaluating trigonometric functions because they simplify to angles with known trigonometric values on the unit circle.

Even and Odd Functions

Trigonometric functions can be categorized into even and odd functions, which helps to simplify calculations.

**Even Functions:**

• An even function satisfies the property \(f(-x) = f(x)\).
• Cosine belongs to this category, meaning \cos(-x) = \cos(x)\.
• This property simplifies calculations, such as when converting negative angles using cosine, since they yield the same result as positive angles.

**Odd Functions:**

• An odd function satisfies \(f(-x) = -f(x)\).
• Sine and tangent are examples of odd functions.
• This property is useful when dealing with negative angles, resulting in negated outcomes compared to their positive counterparts.

The even nature of cosine was used in the exercise to determine \cos(-\frac{7\pi}{4})\ by equating it to \cos(\frac{\pi}{4})\, greatly simplifying the process.