Problem 55 - Precalculus Mathematics for Calculus

Question

Write the expression in terms of sine only. $$-\sqrt{3} \sin x+\cos x$$

Step-by-Step Solution

Answer

The expression is \(-\sqrt{3} \sin x + \sqrt{1-\sin^2 x}\).

Step 1: Identify Trigonometric Identity

The expression given is \(-\sqrt{3} \sin x + \cos x\). The aim is to write it in terms of sine only. We will use the trigonometric identity for cosine:\[ \cos x = \sqrt{1-\sin^2 x} \] to transform \( \cos x \) into a function of \( \sin x \).

Step 2: Substitute Cosine with Sine

Substitute \( \cos x \) in the expression:\[ -\sqrt{3} \sin x + \cos x = -\sqrt{3} \sin x + \sqrt{1-\sin^2 x} \] > Now, the expression is in terms of sine, with both terms containing sine components.

Step 3: Simplify the Expression

The expression\[ -\sqrt{3} \sin x + \sqrt{1-\sin^2 x} \] is already simplified as much as possible in terms of sine, containing both a linear and square-root portion of \( \sin x \). Although not simpler, this is the expression rewritten solely in terms of sine.

Key Concepts

Trigonometric Identities

Trigonometric identities are formulas or equations that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equation are defined. They help in transforming and simplifying trigonometric expressions. A familiar and often-used identity is the Pythagorean identity, which for sine and cosine is given by:\[ \sin^2 x + \cos^2 x = 1 \]From this identity, we can express \( \cos x \) in terms of \( \sin x \) as follows:

• First, rearrange the identity to solve for \( \cos^2 x \): \( \cos^2 x = 1 - \sin^2 x \)
• Take the square root to express \( \cos x \): \( \cos x = \sqrt{1 - \sin^2 x} \)

This transformation is instrumental in converting expressions into a single trigonometric function, which is essential for many trigonometric equation solutions. Recognizing and using trigonometric identities allow you to interconvert and simplify complex expressions during problem-solving.

Expression Simplification

Expression simplification involves the process of translating a complex expression into a simpler or more manageable form. Simplifying trigonometric expressions often involves using identities to rewrite terms in a unified form or reducing terms. This procedure makes it easier to handle mathematical expressions in problem-solving.In the exercise, the original expression is:\[ -\sqrt{3} \sin x + \cos x \]To simplify it, we replace \( \cos x \) using the trigonometric identity:\[ \cos x = \sqrt{1 - \sin^2 x} \]Substituting, we get:\[ -\sqrt{3} \sin x + \sqrt{1 - \sin^2 x} \]This version of the expression is rephrased solely in terms of \( \sin x \), and while it may not appear simplified in the conventional sense of shorter or fewer terms, it now uniformly incorporates one trigonometric function. This simplification helps particularly in certain calculus applications or when solving equations where the aim is to express everything using one function.

Precalculus

Precalculus serves as a bridge between algebra and calculus. It covers various topics that prepare students for calculus, primarily involving the study of functions, trigonometry, and analytic geometry. Mastery of these concepts is crucial for success in calculus. Specific to this exercise, precalculus lays the foundation for understanding and manipulating trigonometric functions like sine and cosine. This understanding is critical for various applications, including:

• Graphing trigonometric functions
• Solving trigonometric equations
• Understanding periodic phenomena

Precalculus enables students to become adept at recognizing when to use particular trigonometric identities, how to transform functions, and how to manage expressions involving angles and triangles. This ability to transform and simplify expressions, as demonstrated in the exercise, is a recurring task in calculus, underscoring the importance of a strong precalculus foundation.