Problem 80 - Algebra and Trigonometry

Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the exact value of \(\sin ^{2} \frac{5 \pi}{8}-\cos ^{2} \frac{5 \pi}{8}\).

Step-by-Step Solution

Answer

\( \frac{\sqrt{2}}{2} \)

Step 1: Recall the trigonometric identity

Recall the trigonometric identity: o \( \sin^2 \theta - \cos^2 \theta = -\cos(2\theta) \)

Step 2: Identify θ

In this problem, \( \theta = \frac{5\pi}{8} \).

Step 3: Apply the identity

Apply the identity to find the value: o \( \sin^2 \frac{5\pi}{8} - \cos^2 \frac{5\pi}{8} = -\cos(2 \times \frac{5\pi}{8}) \).

Step 4: Simplify the argument

Simplify the argument of the cosine function:o \( 2 \times \frac{5\pi}{8} = \frac{10\pi}{8} = \frac{5\pi}{4} \)

Step 5: Evaluate the cosine function

Evaluate the cosine function for the simplified argument:o \( \cos \frac{5\pi}{4} = -\frac{1}{\sqrt{2}} \)

Step 6: Calculate the final value

Substitute the evaluated form back into the equation and simplify: o \( - \cos \frac{5\pi}{4} = - ( - \frac{1}{\sqrt{2}}) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \)

Key Concepts

sine and cosine functions

The sine and cosine functions are fundamental in trigonometry. They are defined based on a right-angled triangle or on the unit circle. For any angle \(\theta\), \(\text{sin} \theta\) corresponds to the y-coordinate, and \(\text{cos} \theta\) corresponds to the x-coordinate of the point on the unit circle.
These functions help us understand patterns related to periodic waves, oscillation, and circular motion. You often see sine and cosine functions in various fields like physics, engineering, and even in music. Understanding these functions is crucial for solving many trigonometric problems, including identities and simplifying expressions.

trigonometric identities

Trigonometric identities are equations that hold true for all values of the variables involved. They are used to simplify complex trigonometric expressions and solve trigonometric equations. One fundamental identity is the Pythagorean identity: \(\text{sin}^2 \theta + \text{cos}^2 \theta = 1\).
In our exercise, we used the identity: \(\text{sin}^2 \theta - \text{cos}^2 \theta = -\text{cos}(2 \theta)\). These identities are derived using the unit circle and are essential for solving trigonometric problems. Knowing these identities can make your work much faster and less cumbersome.

exact values in trigonometry

Exact values in trigonometry refer to the specific values of sine, cosine, tangent, and other trigonometric functions at particular angles. These values are often seen in fractions involving \(\frac{\text{pi}}{4}\), \(\frac{\text{pi}}{3}\), \(\frac{\text{pi}}{6}\), etc.
For instance, \(\text{cos} \frac{5\text{pi}}{4}\) equals -\(\frac{1}{\text{sqrt}(2)}\), which simplifies to \(\frac{-\text{sqrt}(2)}{2}\). Memorizing these values is useful because they frequently pop up in exercises, thus saving you a lot of computation time.

simplifying trigonometric expressions

Simplifying trigonometric expressions often involves using identities and exact values. The goal is to rewrite the expression in its simplest form. This might mean converting all functions to sines and cosines, combining like terms, or using identities.
In our exercise, we started with \(\text{sin}^2 \frac{5\text{pi}}{8} - \text{cos}^2 \frac{5\text{pi}}{8}\). By using the identity \(\text{sin}^2 \theta - \text{cos}^2 \theta = -\text{cos}(2 \theta)\), and then simplifying the argument of cosine, we achieved the final simplified result. Mastering this skill is incredibly useful for solving problems efficiently and accurately.