Problem 16 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-16: Express as a trigonometric function of one angle. $$ \sin (-5) \cos 2+\cos 5 \sin (-2) $$

Step-by-Step Solution

Answer

The expression simplifies to \(-\sin(7)\).

Step 1: Identify the Formula

We are given the expression \( \sin(-5) \cos(2) + \cos(5) \sin(-2) \). This is reminiscent of the sine addition/subtraction formulas, specifically \( \sin(A + B) = \sin A \cos B + \cos A \sin B \).

Step 2: Recognize Negative Angle Identities

We know that \( \sin(-A) = -\sin(A) \) and \( \cos(-A) = \cos(A) \). Apply these identities: \( \sin(-5) = -\sin(5) \) and \( \sin(-2) = -\sin(2) \). The expression becomes \( -\sin(5) \cos(2) - \cos(5) \sin(2) \).

Step 3: Apply the Sine Addition Formula

The expression \( -\sin(5) \cos(2) - \cos(5) \sin(2) \) matches the sine subtraction formula \( -\sin(A + B) = -\sin A \cos B - \cos A \sin B \). Here, \( A = 5 \) and \( B = 2 \). Plug these into the formula: \(-\sin(5+2) = -\sin(7)\).

Step 4: Express the Result

The resulting expression is simply \(-\sin(7)\). This is a single trigonometric function representing the original expression, simplified to one angle.

Key Concepts

Sine Addition Formula

The sine addition formula is a useful tool in trigonometry that helps us simplify expressions involving trigonometric functions of two angles. It states:

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

This formula combines sine and cosine functions of two angles into a single sine function of the sum of those angles. This reduction can simplify calculations and is particularly beneficial in transforming complex expressions into simpler, more manageable forms.
In the original exercise, recognizing the pattern of the sine addition formula helps categorize the expression \( \sin(-5) \cos(2) + \cos(5) \sin(-2) \) as a candidate for simplification using this tool. The formula allows us to rewrite the combination of products as a single sine function of the sum of two angles.

Negative Angle Identities

Negative angle identities play a crucial role in simplifying trigonometric expressions. They describe how sine and cosine behave when the input angle is negative:

• For sine: \( \sin(-A) = -\sin(A) \)
• For cosine: \( \cos(-A) = \cos(A) \)

These identities state that the sine function is an odd function, showing a reflectional symmetry across the origin. The cosine function, meanwhile, is even, maintaining its properties even if the sign of the angle changes.
In our exercise, we applied these identities to the terms \( \sin(-5) \) and \( \sin(-2) \):

• \( \sin(-5) \) becomes \(-\sin(5) \)
• \( \sin(-2) \) becomes \(-\sin(2) \)

This simplification process helps in identifying the structure needed to apply the sine subtraction formula effectively.

Sine Subtraction Formula

The sine subtraction formula is a direct counterpart of the sine addition formula. It helps rewrite a combination of trigonometric expressions as a single sine term for easier computation. It is defined as:

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

This formula is particularly convenient when an original expression closely matches its structure with negative signs. In the original exercise, after simplifying with negative angle identities:

• The expression transformed to \( -\sin(5) \cos(2) - \cos(5) \sin(2) \)

This fits perfectly in the sine subtraction formula's framework, specifically as \( -\sin(5+2) = -\sin(7) \).
Applying this formula, we arrive at the solution \(-\sin(7)\), significantly simplifying the original problem into a well-known trigonometric function of a single angle.