Problem 25 - Precalculus with Calculus Previews

Question

Use a double-angle formula to write the given expression as a single trigonometric function of twice the angle. $$ 1-2 \sin ^{2} \frac{\pi}{5} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\cos\left(\frac{2\pi}{5}\right)\).

Step 1: Identify the Formula

Recognize that the expression \(1 - 2\sin^2\theta\) is related to the double-angle identity for cosine. The formula \(\cos(2\theta) = 1 - 2\sin^2\theta\) can be used to rewrite the given expression as a trigonometric function of twice the angle.

Step 2: Apply the Double-Angle Formula

Apply the double-angle formula for cosine. Substitute \(\theta = \frac{\pi}{5}\) into the formula: \(\cos(2\theta) = 1 - 2\sin^2\theta\). The expression becomes \(\cos(2\cdot \frac{\pi}{5})\).

Step 3: Simplify the Expression

Calculate \(2\cdot\frac{\pi}{5}\) to find the angle for the cosine expression. This results in \(\frac{2\pi}{5}\). Thus, the expression simplifies to \(\cos\left(\frac{2\pi}{5}\right)\).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in understanding relationships in geometry, specifically in triangles. Essential in various mathematical fields, these functions include sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)). Each describes a specific ratio in a right triangle:

• Sine is the ratio of the length of the opposite side to the hypotenuse.
• Cosine is the ratio of the adjacent side to the hypotenuse.
• Tangent is the ratio of the opposite side to the adjacent side.

Understanding these functions and their derivatives is crucial for solving problems involving angles and side lengths. In precalculus, they lay the groundwork for more advanced topics like trigonometric identities and calculus.

Cosine Identity

The cosine identity is one of several trigonometric identities that helps simplify expressions and solve equations. The double-angle identity for cosine—\( \cos(2\theta) = 1 - 2\sin^2\theta \)—is particularly useful. It allows us to express a cosine function as an equivalent term involving sine, providing a different perspective or simplifying complex calculations. By identifying these identities in expressions, one can convert them into more manageable forms, often making calculations easier and revealing hidden relationships within the angles or functions they describe.

Angle Conversion

In trigonometry, converting angles from one form to another helps better analyze problems and simplify computations. Typically, angles can be measured in degrees or radians. For instance, 180 degrees equals \( \pi \) radians. When working with trigonometric identities and functions, it's common to use radians as it often leads to simpler equations.
In our exercise, \( \frac{\pi}{5} \) radians was given, and the problem required finding twice this angle. By multiplying by 2, we converted the original angle to \( \frac{2\pi}{5} \) radians, facilitating the use of the cosine identity in our solution.

Precalculus Concepts

Precalculus is a crucial step before diving into calculus, focusing on functions and their properties. It builds a solid foundation by exploring algebraic skills, functions, and trigonometry deeply. This includes understanding identities like those used in the exercise—the double-angle identity.

• Precalculus helps us grapple with expressions analytically, setting the stage for limits, derivatives, and integrals.
• It emphasizes logical reasoning and the ability to manipulate and transform mathematical expressions.

This understanding is vital for success in calculus, helping to tackle complex mathematical problems with confidence and precision.