Problem 104 - Algebra and Trigonometry

Question

Write each trigonometric expression as an algebraic expression (that is, without any trigonometric functions). Assume that x and y are positive and in the domain of the given inverse trigonometric function. $$ \sin \left(\tan ^{-1} x-\sin ^{-1} y\right) $$

Step-by-Step Solution

Answer

The algebraic expression of the given trigonometric equation \( \sin (\tan^{-1}x - \sin^{-1}y) \) is \( \frac{x\sqrt{1-y^2} - y}{\sqrt{1+x^2}} \)

Step 1 Identify the values

Recognize that the function \( \tan^{-1}x \) is an angle whose tangent is \( x \). Similarly, \( \sin^{-1}y \) is an angle whose sine is \( y \). We will denote these angles as \( A \) and \( B \) respectively.

Step 2 Apply Pythagorean theorem and find cos

For \( A = \tan^{-1}x \), we use the Pythagorean theorem on a right triangle with opposite side \( x \) and adjacent side 1. This gives us \( \cos(A) = \frac{{\text{adjacent side}}}{{\text{hypotenuse}}} = \frac{1}{\sqrt{1+x^2}} \). Similarly, for \( B = \sin^{-1}y \), we find \( \cos(B) = \frac{\sqrt{1-y^2}}{1} = \sqrt{1-y^2} \).

Step 3 Use trigonometric identities for sine of difference of two angles

The expression \( \sin (\tan^{-1}x - \sin^{-1}y) \) is the sine of the difference of two angles. Use the identity \( \sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B) \). This gives \(\sin(\tan^{-1}x)\cos(\sin^{-1}y) - \cos(\tan^{-1}x)\sin(\sin^{-1}y)\). Considering \( \sin(\tan^{-1}x) = x/\sqrt{1+x^2} \) and \( \cos(\sin^{-1}y) = \sqrt{1-y^2} \), and applying these to given equation, which simplifies to \( x\sqrt{1-y^2} - y/\sqrt{1+x^2} \).

Step 4 Simplify the expression

The simplified algebraic expression of given trigonometric equation is \( \frac{x\sqrt{1-y^2} - y}{\sqrt{1+x^2}} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find angles based on given trigonometric values. They reverse the trigonometric functions (sine, cosine, tangent). For example, \( \tan^{-1}(x) \) finds the angle whose tangent is \( x \). Similarly, \( \sin^{-1}(y) \) finds the angle whose sine is \( y \). These functions are essential when converting trigonometric expressions into algebraic ones.
When using inverse trigonometric functions:

• \( \tan^{-1}(x) \) tells us that the opposite side is \( x \), and the adjacent side is 1 in a right triangle. The angle we get from this expression is called \( A \).
• \( \sin^{-1}(y) \) indicates that the opposite side is \( y \) and the hypotenuse is 1, giving us angle \( B \).

These relationships help in translating problems into algebraic forms by understanding the angles and sides involved.

Pythagorean Theorem

The Pythagorean theorem is a vital tool in trigonometry, especially when dealing with inverse functions. The theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This can be written as \( a^2 + b^2 = c^2 \). When working with inverse trigonometric functions:

• For \( A = \tan^{-1}(x) \), we form a triangle where the opposite side is \( x \) and the adjacent side is 1. Using the Pythagorean theorem, the hypotenuse becomes \( \sqrt{1 + x^2} \).
• For \( B = \sin^{-1}(y) \), the opposite side is \( y \) and the hypotenuse is 1. Therefore, the adjacent side is found as \( \sqrt{1 - y^2} \).

This process helps express trigonometric functions in algebraic terms, aiding in solving more complex expressions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are universally true. They help simplify complex trigonometric expressions. One popular identity is the sine of difference formula used in this exercise:

• The identity \( \sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B) \) is used when dealing with expressions like \( \sin(\tan^{-1}x - \sin^{-1}y) \).

By substituting angles \( A \) and \( B \) known from inverse trigonometric functions:

• Find \( \sin(A) \) and \( \cos(A) \) using the triangle from \( \tan^{-1}(x) \).
• Find \( \sin(B) \) and \( \cos(B) \) using the triangle from \( \sin^{-1}(y) \).

Using these identities, we can break down the problem into simpler, algebraic expressions.

Sine of Difference Formula

The sine of difference formula is a specific trigonometric identity used to find the sine of the difference of two angles. It's particularly handy when inverse trigonometric functions are involved. The formula is given by:

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

To apply this formula:

• Identify angles \( A \) and \( B \) using the inverse trigonometric expressions.
• Calculate \( \sin(A) \), \( \cos(B) \), \( \cos(A) \), and \( \sin(B) \) using the sides of the triangles.

This formula helps in transforming the trigonometric expression into an easier-to-handle algebraic form, offering a new perspective to solve complex trigonometric equations. This process highlights the power of a well-chosen identity in simplifying expressions.