Problem 14 - Contemporary Precalculus

Question

Use the half-angle identities to evaluate the given expression exactly. $$\cos \frac{3 \pi}{8}$$

Step-by-Step Solution

Answer

Question: Evaluate the given expression exactly using the half-angle identity for cosine: $$\cos \frac{3\pi}{8}$$ Answer: $$\cos \frac{3\pi}{8} = \sqrt{\frac{1 + \cos \frac{3\pi}{4}}{2}}$$

Step 1: Use half-angle identity

\(\cos\frac{3\pi}{8} = \cos\frac{1}{2}\cdot\frac{3\pi}{4}\). Using \(\cos\frac{\theta}{2} = \pm\sqrt{\frac{1+\cos\theta}{2}}\) with \(\theta = \frac{3\pi}{4}\):

Step 2: Evaluate

\(\cos\frac{3\pi}{4} = -\frac{\sqrt{2}}{2}\). Since \(\frac{3\pi}{8}\) is in Q1 (positive):
\(\cos\frac{3\pi}{8} = \sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{2-\sqrt{2}}{4}} = \frac{\sqrt{2-\sqrt{2}}}{2}\)

Key Concepts

Understanding Half-Angle Identities

Half-angle identities are essential tools in trigonometry that allow us to simplify and solve expressions involving angles that are halves of a more recognizable angle. These identities are derived from the basic trigonometric identities and provide a means to find the exact values of trigonometric functions at certain angles.

For the cosine of a half-angle, the identity is given by:

• \( ext{cos} \left( \frac{x}{2} \right) = \pm \sqrt{\frac{1 + \text{cos} x}{2}} \)

This identity shows that the cosine of a half-angle can be calculated using the cosine of the original angle \(x\).

The choice of the sign (positive or negative) depends on the quadrant in which the angle \(\frac{x}{2}\) lies. This is crucial because the cosine function is positive in the first and fourth quadrants, and negative in the second and third quadrants.

In the example problem, we are tasked with finding \(\cos \frac{3\pi}{8}\). Using half-angle identities, we set \(\frac{x}{2} = \frac{3\pi}{8}\), then solve for \(x = \frac{3\pi}{4}\). Thus, \(\text{cos} \left( \frac{3\pi}{8} \right)\) can be found using the known value of \(\text{cos} \frac{3\pi}{4}\).

Diving Deeper into Cosine

Cosine is one of the primary trigonometric functions used to relate angles in right triangles to the ratios of two sides. Specifically, cosine of an angle \(\theta\) in a right triangle is defined as the adjacent side over the hypotenuse. Formally, it is given by the expression:

• \( ext{cos} \theta = \frac{ ext{adjacent}}{ ext{hypotenuse}}\)

This makes cosine useful in a wide range of applications, from calculating distances to advanced signal processing.

The cosine function is periodic with a period of \(2\pi\), meaning that the cosine of an angle plus \(2\pi\) will yield the same result as the cosine of the angle itself. This cyclic nature is what allows us to use identities like the half-angle identities to evaluate angles that are unconventional or not standard in the unit circle.

In the context of the problem, understanding cosine helps when using the half-angle identities to derive exact trigonometric values like \(\cos \frac{3\pi}{8}\). Remember the key properties of cosine when solving various trigonometric problems.

Calculating Exact Trigonometric Values

Exact trigonometric values are important because they provide precise answers, unlike approximations or decimal values which can be imprecise. They are typically derived using known values from the unit circle or trigonometric identities, such as half-angle or double-angle identities.

When computing these values, it is pivotal to leverage the properties of angles, their quadrants, and the signs of trigonometric functions. The problem, for example, asks for \(\cos \frac{3\pi}{8}\), which requires using the half-angle identity to find:

• \(x = \frac{3\pi}{4}\)
• \( ext{cos} x = -\frac{1}{\sqrt{2}}\), since \(\frac{3\pi}{4}\) lies in the second quadrant where cosine is negative.

Using the half-angle identity, substitute \(\text{cos} \frac{3\pi}{4}\) into the formula to get:

• \(\text{cos} \frac{3\pi}{8} = +\sqrt{\frac{1 + (-\frac{1}{\sqrt{2}})}{2}}\)

Solve to find that \(\cos \frac{3\pi}{8} = \frac{\sqrt{2 + \sqrt{2}}}{2}\).

The mastery of these exact values allows you to solve a range of trigonometric problems with precision and confidence, converting complex or less familiar angles into terms that are familiar and straightforward.