Problem 43 - Algebra and Trigonometry

Question

\(41-44=\) Write the expression in terms of sine only. $$ 5(\sin 2 x-\cos 2 x) $$

Step-by-Step Solution

Answer

The expression simplifies to: \(10\sin x\cos x + 10\sin^2 x - 5\).

Step 1: Express Sine and Cosine using Angle Addition Formula

The problem asks us to express the function in terms of sine only. This typically involves using known trigonometric identities. Our goal is to transform \(\cos 2x\) using elements involving sine. Recall that \(\cos 2x = \cos^2 x - \sin^2 x\), another form is \(1 - 2\sin^2 x\). We will use this second identity to rewrite \(\cos 2x\).

Step 2: Substitute the Identity for Cosine

Substitute the identity into the original expression: \[5(\sin 2x - \cos 2x) = 5(\sin 2x - (1 - 2\sin^2 x)).\]

Step 3: Simplify the Expression

Expand and simplify the expression obtained in Step 2: \[5(\sin 2x - 1 + 2\sin^2 x) = 5\sin 2x - 5 + 10\sin^2 x.\]

Step 4: Final Expression in Terms of Sin

Notice that \(\sin 2x = 2\sin x \cos x\), and while simplifying will insert a sine function, the question was ambiguous initially if finding a complete form of only sine terms separating \(x\) was necessary. For now, the expression closest to further simplification is: \[10\sin x\cos x + 10\sin^2 x - 5.\] Without restricting from elements originally based on multiple identities (\(\sin x, \cos x\)).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental relationships between trigonometric functions that hold true for all angles. These identities are tools that help us simplify complex trigonometric equations and expressions. In this exercise, we utilize the identity for the cosine of a double angle. Specifically, one of the forms of the double angle identity for cosine is \(\cos 2x = 1 - 2\sin^2 x\). This identity expresses \(\cos 2x\) purely in terms of \(\sin x\), allowing us to replace cosine terms with sine, which is crucial for rewriting our expression in terms of sine only.

Some useful trigonometric identities to remember include:

• Double angle identity for sine: \(\sin 2x = 2\sin x \cos x\)
• Pythagorean identity: \(\sin^2 x + \cos^2 x = 1\)
• Reciprocal identities: \(\sin x = \frac{1}{\csc x}\) and \(\cos x = \frac{1}{\sec x}\)

Understanding these identities is key to tackling various trigonometric problems and simplifying expressions effectively.

Angle Addition Formula

The angle addition formulas allow us to find the sine, cosine, or tangent of the sum or difference of two angles. While they are sometimes seen as secondary to basic identities, they provide a deeper layer of understanding to trigonometric transformations. In the current solution process, we didn't directly use the angle addition formula for sine or cosine, but it's closely related to the double angle identities.

For instance, the angle addition formulas state:

• \(\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b\)
• \(\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b\)

These formulas are integral for breaking down or reconstructing expressions when dealing with sums or differences of angles. Specifically, understanding these can help you derive identities for double angles, like our used \(\cos 2x\) identity, by setting \(a = b = x\).

Though not directly applied here, getting familiar with these formulas can significantly streamline solving trigonometric expressions and enable you to connect different trigonometric concepts more coherently.

Sine and Cosine Relationship

The sine and cosine functions are closely interlinked through various identities and properties. Their relationship is not only pivotal for understanding trigonometric functions but is also essential for transforming expressions, like the one in this exercise. In particular, knowing how to express cosine in terms of sine using identities, such as \(\cos 2x = 1 - 2\sin^2 x\), allows for easier manipulation of expressions.

Key concepts to recall include:

• The reciprocal use in identities such as \(\sin^2 x + \cos^2 x = 1\) lets you express \(\cos x\) in terms of \(\sin x\), as \(\cos x = \sqrt{1 - \sin^2 x}\) for positive cosine values.
• The double angle identity that connects sine and cosine: \(\sin 2x = 2\sin x \cos x\).

These relationships are vital when simplifying or re-expressing functions in terms of one trigonometric function, helping to unearth simpler or more insightful forms of complex expressions. Understanding these links enriches your ability to solve trigonometric problems with confidence.