Problem 27 - 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. $$\cos \left(-\frac{\pi}{4}\right)$$

Step-by-Step Solution

Answer

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

Step 1: Recall the Definition of Cosine as an Even Function

A function is even if it satisfies the property \( f(-x) = f(x) \). Cosine is an even function, so \( \cos(-x) = \cos(x) \). This means that the cosine of a negative angle is the same as the cosine of the corresponding positive angle.

Step 2: Apply the Even Function Property to the Problem

Since \( \cos \left(-\frac{\pi}{4}\right) \) needs to be evaluated, apply the property of cosine being even: \( \cos(-\frac{\pi}{4}) = \cos(\frac{\pi}{4}) \).

Step 3: Determine \( \cos(\frac{\pi}{4}) \) Using Unit Circle Knowledge

On the unit circle, \( \frac{\pi}{4} \) radians corresponds to a point where both x and y coordinates are equal, much like a 45-degree angle in a right triangle. Hence, \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \).

Step 4: Conclusion

Thus, \( \cos\left(-\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \).

Key Concepts

Even Function

An even function has a unique characteristic that makes it symmetrical with respect to the y-axis. This means it mirrors itself across this axis. Mathematically, for a function to be even, it must satisfy the equation: \( f(-x) = f(x) \) for all values of \( x \).
In other words, plugging in a negative value of \( x \) gives the same function output as plugging in the positive value of that same \( x \).

• This is why the graph of an even function looks the same on both sides of the y-axis.
• Common examples of even functions include \( x^2 \), \( x^4 \), and of course, the cosine function.

Understanding this property is essential in trigonometry because it helps simplify calculations, such as determining the cosine of negative angles. With cosine being an even function, \( \cos(-x) = \cos(x) \), we can simplify expressions without having to recompute everything.

Cosine Function

The cosine function is one of the primary trigonometric functions that, alongside sine and tangent, helps us understand angles and sides of triangles, as well as circular motion. The cosine of an angle \( \theta \) represents the x-coordinate of the corresponding point on the unit circle.

• For the angle \( \theta = \frac{\pi}{4} \) (or 45 degrees), the unit circle tells us that both \( x \) and \( y \) coordinates are equal.
• This results in the cosine value \( \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \).
• The cosine function repeats its values in a periodic pattern with period \( 2\pi \).

It's useful to remember that because cosine is even, it provides symmetrical values on both sides of the circle, such as \( \cos(-\theta) = \cos(\theta) \). This property allows us to solve for cosine values of negative angles without needing additional calculations.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable(s) for which the functions are defined. They play a crucial role in simplifying and solving trigonometric equations.

• One of the essential identities is the Pythagorean identity: \( \sin^2\theta + \cos^2\theta = 1 \).
• The even property of cosine gives us an identity in itself: \( \cos(-x) = \cos(x) \).
• Other identities include angle sum and difference identities, multiple angle identities, and double angle identities, among others.

Using these identities can help simplify complex trigonometric expressions and solve for unknown angles or lengths in geometric figures. Mastery of these identities will make working with trigonometric functions much more manageable and intuitive.