Problem 15 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-16: Express as a trigonometric function of one angle. $$ \cos 3 \sin (-2)-\cos 2 \sin 3 $$

Step-by-Step Solution

Answer

The expression simplifies to \( -\sin 5 \).

Step 1: Identify the Formula

The problem involves expressing \( \cos 3 \sin (-2) - \cos 2 \sin 3 \)as a single trigonometric function. Recognize that this expression can be rewritten using the sine difference identity.

Step 2: Recall the Sine Difference Formula

Recall the identity for the sine of a difference: \( \sin(A - B) = \sin A \cos B - \cos A \sin B \). Here, it's arranged similarly when comparing it to the given expression.

Step 3: Reapply the Formula

Recognize the structure of the expression:\( \cos 3 \sin (-2) = sin (-2) \cos 3 \text{ and, } -\cos 2 \sin 3 = - \sin 3 \cos 2\).Combine them together to:\( \sin((-2) - 3) \).

Step 4: Simplify the Expression

Simplify the angle in the expression:\( \sin((-2) - 3) = \sin(-5) \).Knowing that \( \sin(-x) = -\sin(x) \), rewrite this as:\( -\sin 5 \).

Key Concepts

Sine Difference Identity

The sine difference identity is a trigonometric identity that helps simplify expressions involving the sine of two angles that are subtracted from each other. This identity states that:

• \( \sin(A - B) = \sin A \cos B - \cos A \sin B \)

This shows how a difference between two angles can be transformed into an expression involving their sine and cosine products. Simplifying expressions using this identity makes it easier to work with them in problems, such as the one involving \( \cos 3 \sin(-2) - \cos 2 \sin 3 \).
In this example, by recognizing the format of the expression, we can group the terms to fit the identity precisely. This leads to expressing the original problem as \( \sin((-2) - 3) \), a simpler form corresponding to the sine of a single angle difference. This not only simplifies the problem but also enhances the efficiency of solving similar trigonometric equations.

Angle Simplification

Angle simplification involves manipulating trigonometric expressions to reduce complex angles to a more manageable form. In problems like the one we're addressing, we'll use angle simplification to turn expressions involving angles such as \((-2) - 3\) into simpler forms.
Here, this process lets us take \( \sin((-2) - 3) \) and convert it into \( \sin(-5) \), a single angle expression. Such simplification is pivotal as it enables easier application of trigonometric identities and rules.
When you face a similar angle involving subtraction like this, always simplify it wherever possible, adhering to the trigonometric rules and identities. The aim is to bring down complexity by dealing with straightforward angles that can further be analyzed using other fundamental identities, leading to efficient problem-solving strategies.

Negative Angle Trigonometry

Negative angle trigonometry deals with trigonometric functions when they involve negative angles. It's crucial to know the behavior of trigonometric functions with negative arguments. For sine, the identity is:

• \( \sin(-x) = -\sin(x) \)

This denotes that sine is an odd function. Thus, when you have an expression such as \( \sin(-5) \), it simplifies to \(-\sin(5)\).
Understanding how negative angles affect trigonometric functions helps in correctly evaluating and simplifying sinusoidal expressions. Proficiency in this area enables tackling problems by simply converting negative angles to positive equivalents, ensuring calculations are exact and solutions straightforward.