Problem 16 - Precalculus Essentials

Question

In Exercises \(15-22,\) write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. $$2 \sin 22.5^{\circ} \cos 22.5^{\circ}$$

Step-by-Step Solution

Answer

The expression \(2 \sin 22.5^{\circ} \cos 22.5^{\circ}\) is equivalent to \(\sin 45^{\circ}\), and the exact value is \(\frac{1}{\sqrt{2}}\) or \(\frac{\sqrt{2}}{2}\).

Step 1: Double-Angle Formula

Applying the double-angle formula for sine, which is \(\sin(2A) = 2 \sin A \cos A\), where \(A=22.5^{\circ}\), given expression can be written as \(\sin(2 \times 22.5^{\circ})\), or \(\sin 45^{\circ}\).

Step 2: Finding Exact Value

The exact value of \(\sin 45^{\circ}\) can be found from the unit circle or special triangles, which is \(\frac{1}{\sqrt{2}}\) or \( \frac{\sqrt{2}}{2}\).

Key Concepts

Double-Angle Formula

The double-angle formula is a fundamental tool in trigonometry, often used for simplifying expressions. It helps in expressing trig functions of double angles, using those of single angles.
For the sine function, the double-angle formula is given by:
\[ \sin(2A) = 2 \sin A \cos A \]

This means, to find the sine of a double angle, we can use the sine and cosine of half that angle. For instance, in the exercise, where \( A = 22.5^{\circ} \), the double-angle formula helps transform the given expression \(2 \sin 22.5^{\circ} \cos 22.5^{\circ}\) into \( \sin 45^{\circ} \), simplifying the computation.

• The formula is essential for trigonometric equations, modeling periodic phenomena, and in calculus, especially in integration and differentiation.
• It allows for transforming and simplifying expressions, making calculations more straightforward and understandable.

Sine Function

The sine function is one of the basic trigonometric functions, which represents periodic oscillations and waves. It is commonly used in various applications including physics, engineering, and even music. For any angle \( \theta \), the sine function calculates the ratio of the length of the opposite side to the hypotenuse in a right triangle.

In terms of unit circle definitions, the sine of an angle is the y-coordinate of the point where the terminal side of the angle intersects the unit circle.

• The sine function varies between -1 and 1, maintaining a wave-like pattern repeated every \( 360^{\circ} \) or \( 2\pi \) radians.
• Important properties of the sine function include its periodicity, symmetry, and its zero crossings at multiples of \( \pi \).
• The sine function is integral to defining wave patterns, oscillations, and is used in Fourier Transform, a critical tool for signal processing.

Exact Values of Trigonometric Functions

Understanding the exact values of trigonometric functions is crucial for solving trig problems accurately without approximations. These values often appear in problems involving specific angles such as \(30^{\circ}\), \(45^{\circ}\), and \(60^{\circ}\).
For example, the sine of \(45^{\circ}\) is often required in calculations.

• The exact value of \( \sin 45^{\circ} \) is \( \frac{\sqrt{2}}{2} \). This comes from recognizing \(45^{\circ}\) as a special angle, representing one-half of a square's diagonal cut.
• These known values come from geometric identities and can be found using methods such as the unit circle approach and special triangles, like 45-45-90 triangles.
• Learning these values by heart is beneficial as they frequently occur and simplify complex problems, reducing reliance on calculators.

When solving trigonometric expressions like the one in the exercise, recognizing these specific trigonometric values allows quick and precise solutions.