Problem 34 - Precalculus with Limits

Question

In Exercises 29-36, use a double-angle formula to rewrite the expression. \( 10 \sin^2 x - 5 \)

Step-by-Step Solution

Answer

The expression \( 10 \sin^2 x - 5 \) can be rewritten as \( -\cos(2x) \) using a double-angle formula.

Step 1: Identify the Given Expression

The given expression is \( 10 \sin^2 x - 5 \).

Step 2: Normalize the Expression

Normalize the expression by dividing the given expression by 5. This gives \( 2 \sin^2 x - 1 \).

Step 3: Use the Double-Angle Formula

Use the double angle formula: \( 2 \sin^2 x = 1 - \cos(2x) \). Instead of \( 2 \sin^2 x \) we can write \( 1 - \cos(2x) \). So, \( 2 \sin^2 x - 1 \) will be \( 1 - \cos(2x) - 1 \).

Step 4: Simplify the Expression

The expression above simplifies to \( -\cos(2x) \).

Key Concepts

Trigonometric Identities

Trigonometric identities are like little shortcuts in mathematics. They allow you to transform and simplify expressions using known formulas. The identity we're focusing on is part of the double angle formulas, which are used to express trigonometric functions at a multiple of an angle. In this exercise, we used:

• Double-angle formula for sine: \( 2 \sin^2 x = 1 - \cos(2x) \)

This formula helps to express \( 2 \sin^2 x \) using cosine, making it easier to work with in many cases. It's all about connecting insights from different trigonometric functions and showing how they relate. Recognizing and using these relationships makes solving trig problems much more manageable.
Start by memorizing these key identities, as they often serve as the foundation for solving more complex trigonometric problems. With practice, you'll find these identities becoming second nature.

Simplifying Trigonometric Expressions

Simplifying expressions is a key skill in mathematics. It involves rewriting an expression in a simpler or more efficient form. This is crucial in trigonometry because simpler forms are easier to evaluate and manipulate.

• Begin by identifying parts of the expression that can match known formulas or identities.
• Substitute those parts with the equivalent identity, as we did with \( 2 \sin^2 x = 1 - \cos(2x) \).
• Simplify algebraically to reach an expression that's efficient for computation. For example, from \( 2 \sin^2 x - 1 \) to \( -\cos(2x) \).

This process can not only make solving an equation simpler but also more insightful. It's like peeling away layers of complexity to reveal a more streamlined solution strategy.

Precalculus

Precalculus lays the groundwork for calculus by strengthening algebra and trigonometry skills. One of the main goals is to develop a comprehensive understanding of functions and equations, including trigonometric functions.

• It introduces you to identities and formulas that are frequently used in calculus.
• Precalculus emphasizes the manipulation and transformation of different mathematical expressions, like rewriting trigonometric expressions using identities to simplify them.
• This practice helps build strong analytical skills, which are necessary for success in calculus.

Through exercises like applying the double angle formula, you start building a toolkit of techniques that prepare you for more advanced mathematics. Understanding these concepts enriches your mathematical insight and equips you with versatile problem-solving strategies.