Incorrectly substituting expressions

Oct 13, 2025 by windailys.com 53 views

Introduction

In this article, we will explore the relationship between trigonometric functions and their expressions in terms of given values. We will start by expressing sin 2x + sin 2y over sin 2x - sin 2y in terms of a and b, and then use this result to obtain an expression for tan (x + 3y) in terms of a and b.

Given Equations

We are given two equations:

(x + y) Close equals to a
tan x - y = b

Our goal is to express sin 2x + sin 2y over sin 2x - sin 2y in terms of a and b, and then use this result to obtain an expression for tan (x + 3y) in terms of a and b.

Expressing sin 2x + sin 2y over sin 2x - sin 2y in Terms of a and b

To express sin 2x + sin 2y over sin 2x - sin 2y in terms of a and b, we can start by using the given equations to express x and y in terms of a and b.

From equation 1, we can express x + y as a:

x + y = a

We can then express x and y in terms of a as:

x = a - y y = a - x

Substituting these expressions into the equation for tan x - y = b, we get:

tan (a - y) - (a - x) = b

Using the identity tan (a - b) = (tan a - tan b) / (1 + tan a tan b), we can rewrite the equation as:

(tan a - tan y) / (1 + tan a tan y) - (a - x) = b

Simplifying the equation, we get:

tan a - tan y - (a - x) = b (1 + tan a tan y)

We can then express tan y in terms of a and b as:

tan y = (a - x) - b (1 + tan a tan y)

Substituting this expression into the equation for sin 2x + sin 2y over sin 2x - sin 2y, we get:

(sin 2x + sin 2y) / (sin 2x - sin 2y) = (2 sin x cos x + 2 sin y cos y) / (2 sin x cos x - 2 sin y cos y)

Using the identity sin 2x = 2 sin x cos x, we can rewrite the equation as:

(sin 2x + sin 2y) / (sin 2x - sin 2y) = (sin 2x + sin 2y) / (sin 2x - sin 2y)

Simplifying the equation, we get:

(sin 2x + sin 2y) / (sin 2x - sin 2y) = 1 + tan a tan y

Substituting the expression for tan y in terms of a and b, we get:

(sin 2x + sin 2y) / (sin 2x - sin 2y) = 1 + ((a - x) - b (1 + tan a tan y)) tan a

Simplifying the equation, we get:

(sin 2x + sin 2y) / (sin 2x - sin 2y) = 1 + (a - x) tan a - b (1 + tan a tan y) tan a

Expressing tan (x + 3y) in Terms of a and b

To express tan (x + 3y) in terms of a and b, we can use the result we obtained earlier for sin 2x + sin 2y over sin 2x - sin 2y in terms of a and b.

We can start by expressing tan (x + 3y) in terms of tan x and tan y as:

tan (x + 3y) = (tan x + 3 tan y) / (1 - 3 tan x tan y)

Using the identity tan (a + b) = (tan a + tan b) / (1 - tan a tan b), we can rewrite the equation as:

tan (x + 3y) = (tan x + 3 tan y) / (1 - 3 tan x tan y)

We can then express tan x and tan y in terms of a and b using the given equations.

From equation 1, we can express x + y as a:

x + y = a

We can then express x and y in terms of a as:

x = a - y y = a - x

Substituting these expressions into the equation for tan x - y = b, we get:

tan (a - y) - (a - x) = b

Using the identity tan (a - b) = (tan a - tan b) / (1 + tan a tan b), we can rewrite the equation as:

(tan a - tan y) / (1 + tan a tan y) - (a - x) = b

Simplifying the equation, we get:

tan a - tan y - (a - x) = b (1 + tan a tan y)

We can then express tan y in terms of a and b as:

tan y = (a - x) - b (1 + tan a tan y)

Substituting this expression into the equation for tan (x + 3y) in terms of tan x and tan y, we get:

tan (x + 3y) = (tan x + 3 ((a - x) - b (1 + tan a tan y))) / (1 - 3 tan x ((a - x) - b (1 + tan a tan y)))

Simplifying the equation, we get:

tan (x + 3y) = (tan x + 3 (a - x) - 3 b (1 + tan a tan y)) / (1 - 3 tan x (a - x) + 3 b tan x (1 + tan a tan y))

Conclusion

In this article, we have expressed sin 2x + sin 2y over sin 2x - sin 2y in terms of a and b, and then used this result to obtain an expression for tan (x + 3y) in terms of a and b. We have shown that tan (x + 3y) can be expressed in terms of a and b as:

tan (x + 3y) = (tan x + 3 (a - x) - 3 b (1 + tan a tan y)) / (1 - 3 tan x (a - x) + 3 b tan x (1 + tan a tan y))

Q: What is the relationship between the given equations and the expression for sin 2x + sin 2y over sin 2x - sin 2y?

A: The given equations (x + y) Close equals to a and tan x - y = b are used to express x and y in terms of a and b. These expressions are then substituted into the equation for sin 2x + sin 2y over sin 2x - sin 2y to obtain the final expression in terms of a and b.

Q: How is the expression for tan (x + 3y) in terms of a and b obtained?

A: The expression for tan (x + 3y) in terms of a and b is obtained by using the result for sin 2x + sin 2y over sin 2x - sin 2y in terms of a and b. The equation for tan (x + 3y) is then expressed in terms of tan x and tan y, and the expressions for tan x and tan y are substituted in terms of a and b.

Q: What is the significance of the given equations in the context of the problem?

A: The given equations (x + y) Close equals to a and tan x - y = b provide a relationship between the variables x and y in terms of the constants a and b. This relationship is used to express x and y in terms of a and b, which is then used to obtain the final expressions for sin 2x + sin 2y over sin 2x - sin 2y and tan (x + 3y) in terms of a and b.

Q: How can the result for sin 2x + sin 2y over sin 2x - sin 2y in terms of a and b be used in real-world applications?

A: The result for sin 2x + sin 2y over sin 2x - sin 2y in terms of a and b can be used in various real-world applications, such as:

Engineering: The result can be used to design and analyze mechanical systems, such as gears and linkages, where the relationship between the angles of rotation is critical.
Physics: The result can be used to describe the motion of objects in terms of their angular displacement and velocity.
Computer Science: The result can be used in computer graphics and game development to create realistic animations and simulations.

Q: What are some common mistakes to avoid when working with trigonometric functions and their expressions in terms of given values?

A: Some common mistakes to avoid when working with trigonometric functions and their expressions in terms of given values include:

Incorrectly substituting expressions: Make sure to substitute the correct expressions for the variables in terms of the given values.
Failing to simplify expressions: Make sure to simplify the expressions as much as possible to obtain the final result.
Not checking for domain restrictions: Make sure to check for any domain restrictions on the trigonometric functions and their expressions in terms of given values.

Q: How can the result for tan (x + 3y) in terms of a and b be used in real-world applications?

A: The result for tan (x + 3y) in terms of a and b can be used in various real-world applications, such as:

Navigation: The result can be used to determine the direction of a ship or aircraft in terms of its angular displacement and velocity.
Surveying: The result can be used to determine the orientation of a surveyor's instrument in terms of its angular displacement and velocity.
Computer Science: The result can be used in computer graphics and game development to create realistic animations and simulations.