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Chapter 1: Problem 36

Use the addition formulas to derive the identities. $$\sin (A-B)=\sin A \cos B-\cos A \sin B$$

Short Answer

Expert verified
\( \sin(A-B) = \sin A \cos B - \cos A \sin B \) is derived using the addition identity and even-odd properties.

Step by step solution

01

Recall the Definition of Sine Addition Identity

The sine addition formula is given by \( \sin(A+B) = \sin A \cos B + \cos A \sin B \). We will use this identity to derive the sine subtraction formula.
02

Express \( \\sin(A-B) \) Using Addition Identity

We know that \( \sin(A-B) = \sin(A + (-B)) \). Using the sine addition identity, this becomes \( \sin(A) \cos (-B) + \cos(A) \sin(-B) \).
03

Apply Trigonometric Even-Odd Properties

The cosine function is even, so \( \cos(-B) = \cos B \), and the sine function is odd, so \( \sin(-B) = -\sin B \). Therefore, substitute into the equation: \( \sin(A-B) = \sin A \cos B - \cos A \sin B \).
04

Conclude the Sine Subtraction Identity

By using the addition formula and the properties of sine and cosine, we have derived the identity \( \sin(A-B) = \sin A \cos B - \cos A \sin B \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Addition Formulas
Understanding addition formulas is crucial for mastering trigonometry. These formulas relate the sine and cosine functions when two angles are added together. The general addition formulas are:
  • \( \sin(A+B) = \sin A \cos B + \cos A \sin B \)
  • \( \cos(A+B) = \cos A \cos B - \sin A \sin B \)
These allow us to express the trigonometric function of the sum of two angles in terms of the functions of the individual angles.
Addition formulas are incredibly useful in many real-world applications, including signal processing and solving complex trigonometric equations. By learning how these formulas are derived and applied, learners can deepen their understanding of trigonometry and widen their problem-solving capabilities.
Sine Subtraction
The concept of subtraction in trigonometry can be initially confusing, but it's closely related to addition formulas. For sine subtraction, you can think of it as adding a negative angle.
This is expressed as: \( \sin(A-B) = \sin(A + (-B)) \). From the sine addition formula \( \sin(A+B) = \sin A \cos B + \cos A \sin B \), we apply the same idea by treating \(-B\) as a negative angle.
With the trigonometric properties of
  • \( \cos(-B) = \cos B \)
  • \( \sin(-B) = -\sin B \)
we derive that:
\( \sin(A-B) = \sin A \cos B - \cos A \sin B \).
This identity is vital for simplifying expressions and solving equations involving sine of angle differences.
Even-Odd Properties
Sine and cosine functions have interesting symmetrical properties called even and odd properties, which are pivotal in understanding trigonometric formulas.
A function is even if \( f(-x) = f(x) \) and odd if \( f(-x) = -f(x) \).
  • Cosine is an even function: \( \cos(-\theta) = \cos(\theta) \) because its graph is symmetrical about the y-axis.
  • Sine is an odd function: \( \sin(-\theta) = -\sin(\theta) \) because its graph is symmetrical about the origin.
These properties help simplify trigonometric expressions significantly.
For example, in deriving the sine subtraction formula, these properties allow us to accurately express \( \cos(-B) \) as \( \cos B \) and \( \sin(-B) \) as \(-\sin B \), underscoring the precision these identities bring to trigonometry.

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Most popular questions from this chapter

Tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph. \(y=1-x^{3},\) stretched horizontally by a factor of 2.

Tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph. \(y=1+\frac{1}{x^{2}}, \quad\) compressed vertically by a factor of 2.

Assume that \(f\) is an even function, \(g\) is an odd function, and both \(f\) and \(g\) are defined on the entire real line \((-\infty, \infty) .\) Which of the following (where defined) are even? odd? foLLowing a. \(f g\) b. \(f / g\) c. \(g / f\) d. \(f^{2}=f f\) e. \(g^{2}=g g\) f. \(f^{\circ} g\) g. \(g \circ f\) h. \(f \circ f\) i. \(g^{\circ} \mathrm{g}\)

The variables \(r\) and \(s\) are inversely proportional, and \(r=6\) when \(s=4 .\) Determine \(s\) when \(r=10.\)

Graph the functions. $$y=\frac{1}{x-2}.$$
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