Problem 4 - A Graphical Approach to Precalculus with Limits

Question

Match each expression in Column I with its value in Column \(I I\). A. \(\frac{1}{2}\) B. \(\frac{\sqrt{2}}{2}\) c. \(\frac{\sqrt{3}}{2}\) D. \(-\sqrt{3}\) E. \(\frac{\sqrt{3}}{3}\) F. \(\sqrt{3}\) $$\cos ^{2} \frac{\pi}{6}-\sin ^{2} \frac{\pi}{6}$$

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{1}{2} \), matching A.

Step 1: Identify the Expression

We are given the expression \( \cos^2 \frac{\pi}{6} - \sin^2 \frac{\pi}{6} \). We need to evaluate this to find which value from Column II it matches.

Step 2: Calculate \( \cos \frac{\pi}{6} \)

The cosine of \( \frac{\pi}{6} \) is a trigonometric identity. From the unit circle, we know \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \).

Step 3: Square the Cosine Value

Square the value of cosine:\[\cos^2 \frac{\pi}{6} = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}\]

Step 4: Calculate \( \sin \frac{\pi}{6} \)

From the unit circle, we know \( \sin \frac{\pi}{6} = \frac{1}{2} \).

Step 5: Square the Sine Value

Square the value of sine:\[\sin^2 \frac{\pi}{6} = \left(\frac{1}{2}\right)^2 = \frac{1}{4}\]

Step 6: Substitute and Simplify

Substitute the squared values into the expression:\[\cos^2 \frac{\pi}{6} - \sin^2 \frac{\pi}{6} = \frac{3}{4} - \frac{1}{4} = \frac{2}{4} = \frac{1}{2}\]

Step 7: Match the Result with Column II

The value \( \frac{1}{2} \) from our calculations matches expression A. in Column I.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry, providing a simple way to understand trigonometric functions at different angles. It is a circle with a radius of 1 centered at the origin of a coordinate plane. The unit circle allows us to define the sine and cosine functions for different angles based on their coordinates.

• Angles on the unit circle are measured in radians, a common unit in trigonometry. One full rotation around the circle is equal to \(2\pi\) radians, or 360 degrees.
• Each point on the unit circle can be represented by coordinates \((x, y)\), where \(x\) is the value of the cosine function and \(y\) is the value of the sine function for that angle.
• A crucial trigonometric identity derived from the unit circle is \(\cos^2 \theta + \sin^2 \theta = 1\) for any angle \(\theta\).

Understanding the unit circle helps determine the sine and cosine of angles in standard positions, like \(\frac{\pi}{6}\), and allows for easy calculation and visualization of these values in trigonometry.

Cosine Function

The cosine function is a fundamental trigonometric function, which assigns to any given angle the x-coordinate of the corresponding point on the unit circle. In our exercise, we focus on finding \(\cos \frac{\pi}{6}\).

• To solve trigonometric expressions, such as \(\cos^2 \frac{\pi}{6}\), it is helpful to recall the known identities from the unit circle.
• According to the unit circle, \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\).
• This value can be squared to find \(\cos^2 \frac{\pi}{6} = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}\).

Notably, understanding how this function works helps simplify complex trigonometric expressions and solve related mathematical problems effectively.

Sine Function

The sine function, another essential trigonometric function, determines the y-coordinate of a point on the unit circle for a given angle. In our exercise, it specifically involves finding \(\sin \frac{\pi}{6}\).

• From the unit circle, we know \(\sin \frac{\pi}{6} = \frac{1}{2}\).
• Squaring this value, we obtain \(\sin^2 \frac{\pi}{6} = \left(\frac{1}{2}\right)^2 = \frac{1}{4}\).

The sine function is crucial in calculations involving trigonometric expressions. Understanding the sine function enables us to calculate related values and apply them to solve problems, such as the expression \(\cos^2 \frac{\pi}{6} - \sin^2 \frac{\pi}{6}\).