Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find identities analogous to the addition identities for each expression. (a) \(\sin (x-y)\) (b) \(\cos (x-y)\) (c) \(\tan (x-y)\)

Step-by-Step Solution

Answer

These identities are the subtraction formulas for sine, cosine, and tangent: \(\sin(x-y)\), \(\cos(x-y)\), and \(\tan(x-y)\).

Step 1: Understand the problem

We are tasked with finding identities similar to the addition identities, but for subtraction instead. Specifically, we need to derive formulas for \( \sin(x-y) \), \( \cos(x-y) \), and \( \tan(x-y) \).

Step 2: Recall addition identities

For the addition identities, we have:1. \( \sin(a+b) = \sin a \cos b + \cos a \sin b \)2. \( \cos(a+b) = \cos a \cos b - \sin a \sin b \)3. \( \tan(a+b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \)

Step 3: Derive subtraction identity for \(\sin(x-y)\)

Using similar logic and the angle subtraction formula, we derive:\[ \sin(x-y) = \sin x \cos y - \cos x \sin y \]

Step 4: Derive subtraction identity for \(\cos(x-y)\)

By employing the cosine angle subtraction formula, we find:\[ \cos(x-y) = \cos x \cos y + \sin x \sin y \]

Step 5: Derive subtraction identity for \(\tan(x-y)\)

For the tangent subtraction formula, we use:\[ \tan(x-y) = \frac{\tan x - \tan y}{1 + \tan x \tan y} \]

Step 6: Summarize identities

The identities for subtraction are:1. \( \sin(x-y) = \sin x \cos y - \cos x \sin y \)2. \( \cos(x-y) = \cos x \cos y + \sin x \sin y \)3. \( \tan(x-y) = \frac{\tan x - \tan y}{1 + \tan x \tan y} \)

Key Concepts

Angle Subtraction Formulas

Angle subtraction formulas allow us to find the trigonometric values of the difference of two angles. Using these formulas, we can derive identities that help simplify expressions involving subtraction of angles.

For sine, the subtraction formula is:

• \[ \sin(x-y) = \sin x \cos y - \cos x \sin y \]

This formula shows us how to express the sine of the difference of two angles \((x-y)\) in terms of the sines and cosines of the individual angles.

The cosine subtraction formula works similarly:

• \[ \cos(x-y) = \cos x \cos y + \sin x \sin y \]

Here, the cosine of the difference \((x-y)\) is shown as a combination of the cosines and sines of each angle.

For tangent, the subtraction formula is slightly different:

• \[ \tan(x-y) = \frac{\tan x - \tan y}{1 + \tan x \tan y} \]

These formulas are essential tools for solving problems involving trigonometric functions and help in various mathematical applications.

Sine, Cosine, and Tangent Formulas

The sine, cosine, and tangent functions are foundational elements of trigonometry. They are based on the ratios of sides in a right triangle and apply to any angle using the unit circle.

The sine function, \( \sin(x) \), gives us the ratio of the opposite side to the hypotenuse in a right triangle.
The cosine function, \( \cos(x) \), is the ratio of the adjacent side to the hypotenuse.

The tangent function is the ratio of the sine to the cosine of an angle, or equivalently, the opposite side to the adjacent side:

• \[ \tan(x) = \frac{\sin(x)}{\cos(x)} \]

These functions allow us to calculate angles and distances in various contexts, from simple geometric problems to complex real-world applications.

Understanding how these functions work and how they relate through identity formulas is crucial in trigonometry.

Trigonometric Addition Identities

Trigonometric addition identities simplify the process of finding the trigonometric values of the sum of two angles. These identities are incredibly useful in both theoretical and practical applications of trigonometry.

For the sine of a sum, the identity is:

• \[ \sin(a+b) = \sin a \cos b + \cos a \sin b \]

This formula tells us how to combine the sines and cosines of two angles \(a\) and \(b\) to find the sine of their sum.

The cosine addition identity is:

• \[ \cos(a+b) = \cos a \cos b - \sin a \sin b \]

This identity shows us how to calculate the cosine of the sum of two angles in terms of their individual sines and cosines.

For tangent, the addition identity is expressed as:

• \[ \tan(a+b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]

Mastering these identities allows us to break down complex expressions and solve trigonometric equations more easily.