Problem 3 - 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. \(\sqrt{3}\) $$2 \sin 22.5^{\circ} \cos 22.5^{\circ}$$

Step-by-Step Solution

Answer

The correct match for the expression is option B: \(\frac{\sqrt{2}}{2}\).

Step 1: Identify the Formula

We start by recognizing that the expression we need to evaluate is related to a well-known trigonometric identity. The given expression is \(2 \sin \theta \cos \theta\). We know from trigonometric identities that \(2 \sin \theta \cos \theta = \sin(2\theta)\).

Step 2: Calculate the Angle for the Identity

Given the expression \(2 \sin 22.5^{\circ} \cos 22.5^{\circ}\), we substitute \(22.5^\circ\) for \(\theta\). Thus, the angle used in the double angle formula becomes \(2 \times 22.5^\circ = 45^\circ\).

Step 3: Evaluate the Sine of the Calculated Angle

According to the trigonometric identity and the calculation in Step 2, \(\sin(45^\circ)\) is what we need to compute. From fundamental trigonometric values, we know that \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\).

Step 4: Match the Result with Column II

Now, match the result from Step 3, which is \(\frac{\sqrt{2}}{2}\), with the given values in Column II. This corresponds to option B in Column I.

Key Concepts

Double Angle Formula

The Double Angle Formula is a key concept in trigonometry that simplifies expressions involving sine and cosine. This formula is especially helpful when dealing with angles that appear as twice another angle, as it allows for substitution and simplification. The specific double angle formula for sine is \( \sin(2\theta) = 2\sin\theta\cos\theta \). This means that when you have an expression like \( 2\sin\theta\cos\theta \), it can be rewritten as \( \sin(2\theta) \). This is useful when simplifying expressions or solving equations that involve sine and cosine of one angle, and you want to express it in terms of double the angle. To apply the double angle formula, simply identify your original angle \( \theta \), and then calculate \( 2\theta \).

• Reduces complexity by converting products
• Efficient for solving trigonometric equations

In our exercise, this formula helped us transform \( 2\sin{22.5^\circ}\cos{22.5^\circ} \) into \( \sin(45^\circ) \), simplifying the calculation significantly.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are fundamental concepts in mathematics, especially when dealing with angles and periodic phenomena. They are periodic functions, meaning they repeat their values in regular intervals.

• \( \sin(\theta) \) represents the vertical component of the angle in the unit circle.
• \( \cos(\theta) \) represents the horizontal component.

These functions are defined on the unit circle, where the circle has a radius of 1. Thus, understanding the basic values and relationships of these functions is crucial for solving many trigonometric problems. In our exercise, the expression \( 2\sin(22.5^\circ)\cos(22.5^\circ) \) relies on the double angle identity to calculate the equivalent sine value for a double angle. By knowing that the sine of 45 degrees is \( \frac{\sqrt{2}}{2} \), we could match this value to the given options in the problem set.

Angle Calculation

Angle calculation in trigonometry often involves using various identities and rules to convert and evaluate angles, making calculations simpler and more efficient. Knowing the relationships between angles and how trigonometric functions react to these angles is essential. For the problem given, we started with an angle of \( 22.5^\circ \). By using the double angle formula, we doubled the angle to obtain \( 45^\circ \). Here is why angle calculation matters:

• Allows use of known trigonometric values for ease of calculation
• Often involves critical analysis of expressions using fractions of pi (like \( \pi/4 \) for 45 degrees)

The key is to use precise calculation to align with accurate trigonometric values, which are typically memorized as part of mathematical education. These calculations are instrumental in defining complex periodic movements or evaluations within mathematical models.