Problem 19 - A Graphical Approach to Precalculus with Limits

Question

Use an identity to write each expression as a single trigonometric function or as a single number in exact form. Do not use a calculator. $$1-2 \sin ^{2} 15^{\circ}$$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{\sqrt{3}}{2} \).

Step 1: Recognize the Identity

The expression \(1 - 2 \sin^2 \theta\) can be recognized as a trigonometric identity. This identity is a form of the double angle identity for cosine: \(\cos(2\theta) = 1 - 2\sin^2(\theta)\).

Step 2: Apply the Double Angle Identity

Given the expression \(1 - 2 \sin^2 15^{\circ}\), we recognize this as \(\cos(2 \times 15^{\circ})\). This simplifies to \(\cos(30^{\circ})\).

Step 3: Calculate the Exact Value of Cosine

The cosine of 30 degrees is a common value in trigonometry. It is known that \(\cos(30^{\circ}) = \frac{\sqrt{3}}{2}\).

Key Concepts

Double Angle Identity

The double angle identity is a crucial trigonometric formula that helps simplify expressions involving angles. It's particularly useful when dealing with terms like \(1 - 2\sin^2\theta\), as it can transform them into a more manageable form. In this context, the double angle identity of cosine is especially handy. The identity states that \(\cos(2\theta) = 1 - 2\sin^2(\theta)\). By recognizing this identity, we can reframe our expressions involving sine squared into a simpler cosine expression. This makes calculating specific angle values more straightforward, and by understanding this identity, it becomes easier to navigate through various trigonometric problems.

Remember, identities like the double angle identity not only simplify calculations but also reveal connections between different trigonometric functions. Such insights are invaluable, making problem-solving in trigonometry less daunting.

Cosine Function

The cosine function is a fundamental part of trigonometry. It's a periodic function that describes how angles relate to the ratios of the sides in right triangles. Through the unit circle approach, the cosine of an angle represents the x-coordinate of a point on the circle's circumference. As a key player in the trigonometric world, cosine allows us to transition smoothly between angles and their corresponding numeric values.

In our exercise, we worked with the cosine of twice the angle, \(\cos(30^{\circ})\), derived using the double angle identity. Cosine helps to decode angles into numbers, making trigonometry tangible and less abstract. When dealing with angles like 30 degrees, understanding how cosine operates can help in determining exact trigonometric values, which are essential in calculations involving geometry, physics, and engineering.

• It's important to remember the key values of cosine for special angles like 0°, 30°, 45°, 60°, and 90°.
• Knowing these values empowers you to tackle more complicated problems without always having to revert to a calculator.

Exact Trigonometric Values

Exact trigonometric values are specific values for trigonometric functions at particular angles, typically taught early in trigonometry studies. These values, such as \(\cos(30^{\circ}) = \frac{\sqrt{3}}{2}\), are not just approximations but precise numbers that appear frequently in mathematical problems. Having this knowledge at your fingertips allows you to quickly solve trigonometric problems that involve common angles.

Such values show the importance of memorization in trigonometry—knowing them can significantly speed up your problem-solving process.

• They play an essential role in transforming complex trigonometric expressions into simple numbers.
• This is crucial for exercises requiring exact answers, where any approximation would be insufficient.

Mastering these exact values not only aids in solving textbook exercises but also prepares you for their practical applications in various scientific fields.