Problem 46 - Algebra and Trigonometry

Question

Write the given expression in terms of x and y only. $$ \sin \left(\sin ^{-1} x+\cos ^{-1} y\right) $$

Step-by-Step Solution

Answer

The expression is \( x \cdot y + \sqrt{1-x^2} \cdot \sqrt{1-y^2} \).

Step 1: Understanding the Inverse Trigonometric Function Properties

Recall the properties of inverse trigonometric functions: 1. \( \sin^{-1} x \) gives an angle \( \theta \) such that \( \sin \theta = x \).2. \( \cos^{-1} y \) gives an angle \( \phi \) such that \( \cos \phi = y \). The sum \( \sin^{-1} x + \cos^{-1} y \) involves adding these two angles.

Step 2: Analyze the Identity for Sum of Angles

Consider the trigonometric identity for the sine of the sum of two angles: \[\sin(a + b) = \sin a \cos b + \cos a \sin b\]Here, let \( a = \sin^{-1} x \) and \( b = \cos^{-1} y \). The goal is to express \( \sin(a + b) \) in terms of \( x \) and \( y \).

Step 3: Express Sine and Cosine of Each Angle

Using the definitions from Step 1:- For \( a = \sin^{-1} x \), we have \( \sin a = x \) and by Pythagorean identity, \( \cos a = \sqrt{1-x^2} \).- For \( b = \cos^{-1} y \), we have \( \cos b = y \) and by Pythagorean identity, \( \sin b = \sqrt{1-y^2} \).

Step 4: Substitute in the Identity

Substitute the expressions for sine and cosine from Step 3 into the identity from Step 2:\[\sin(\sin^{-1} x + \cos^{-1} y) = x \cdot y + \sqrt{1-x^2} \cdot \sqrt{1-y^2}\]This expression is now written exclusively in terms of \( x \) and \( y \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are the tools used to find angles when the trigonometric ratio is known. They are the inverses of the standard trigonometric functions such as sine, cosine, and tangent.

- The function \( \sin^{-1} x \) or arcsine is used to find an angle whose sine value is \( x \).
- Similarly, the function \( \cos^{-1} y \) or arccosine finds an angle whose cosine value is \( y \).These functions are critical because they allow us to navigate between ratios and angles. In problems like the one above, they help translate known mathematical relationships into angles that can be manipulated using trigonometric identities.

Sum of Angles Identity

The sum of angles identity is a pivotal formula in trigonometry that helps calculate the result of adding two angles. It's especially useful when these angles are expressions of inverse trigonometric functions.

The identity for sine of a sum is:

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

This formula breaks down the complex expression \( \sin(a + b) \) into more manageable components involving individual sines and cosines.

In the given exercise, this identity is applied by setting \( a = \sin^{-1} x \) and \( b = \cos^{-1} y \). By expressing the sum of these angles using the identity, it becomes possible to substitute known values and simplify the expression.

Pythagorean Identity

The Pythagorean identity is a fundamental trigonometric identity that relates the square of the sine and cosine of an angle to one. It is derived from the Pythagorean theorem for a unit circle.

The identity states:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

For the exercise, this identity is used to find the complementary trigonometric function when the inverse function returns one of them. For instance:

• If \( \sin a = x \), then \( \cos a = \sqrt{1-x^2} \).
• Similarly, if \( \cos b = y \), then \( \sin b = \sqrt{1-y^2} \).

This identity helps express the sine and cosine of the inverse angles in terms of \( x \) and \( y \), which are crucial to finalizing the solution by creating a completely variable-only expression.