Problem 16 - Precalculus: Functions and Graphs

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

It's equivalent to \(-\sin(3)\).

Step 1: Identify the Trigonometric Identity

The expression \( \sin(-5) \cos(2) + \cos(5) \sin(-2) \) resembles the sine addition formula \( \sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B) \). Here, \( A = -5 \) and \( B = 2 \).

Step 2: Apply the Sine Addition Formula

Using the sine addition formula, we substitute \( A = -5 \) and \( B = 2 \) into \( \sin(A + B) \). This gives us \( \sin(-5 + 2) \).

Step 3: Simplify the Expression

Simplify \( \sin(-5 + 2) \) by calculating \( -5 + 2 = -3 \). Thus, the expression simplifies to \( \sin(-3) \).

Step 4: Use the Odd Function Property of Sine

Since sine is an odd function, \( \sin(-x) = -\sin(x) \). Therefore, \( \sin(-3) = -\sin(3) \).

Key Concepts

Sine Addition Formula

When working with trigonometric identities, one important formula is the *sine addition formula*. This formula is used to simplify expressions where the sum of angles is involved. The formula states: \[ \sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B) \]This can be incredibly useful because it helps to combine multiple angle terms into a single sine term. In our specific exercise, we encountered the expression \(\sin(-5) \cos(2) + \cos(5) \sin(-2)\). This structure indicates that we can apply the sine addition formula. - By recognizing this form, we identify the corresponding angles, \(A = -5\) and \(B = 2\), and apply the formula to get \(\sin(-5 + 2)\). The sine addition formula simplifies computations and is a fundamental tool in trigonometry when dealing with multiple-angle expressions.

Trigonometric Simplification

Once we have applied the appropriate identity, the next step is *trigonometric simplification*. This involves resolving the expression into its simplest form. After using the sine addition formula, we arrive at \(\sin(-5+2)\). - Simplifying the expression \(-5+2\) results in \(-3\).At this point, our trigonometric expression converts to \(\sin(-3)\). Simplifying mathematical expressions is crucial in obtaining an answer in its neatest form, which is often necessary for deeper analysis or further calculations.Make sure to handle arithmetic operations carefully, as errors here can lead to incorrect results in later steps.

Odd Function Property

In trigonometry, understanding the properties of functions helps in further simplification and understanding of expressions. One such important property is the *odd function property* of the sine function.- A function \(f(x)\) is considered odd if it satisfies the property \(f(-x) = -f(x)\).For the sine function, this property is applicable: \(\sin(-x) = -\sin(x)\). This means that the graph of sine is symmetric about the origin.In our simplified expression \(\sin(-3)\), we use this property to transform it into \(-\sin(3)\). Understanding and applying these properties allow for more streamlined calculations, ensuring that trigonometric expressions are expressed in a straightforward and manageable manner.