Problem 103 - 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. $$ \cos \left(\sin ^{-1} x-\cos ^{-1} y\right) $$

Step-by-Step Solution

Answer

The algebraic equivalent of the trigonometric expression \(\cos (\sin^{-1} x - \cos^{-1} y)\) is \(\sqrt{1 - x^2} * y + x * \sqrt{1 - y^2}\)

Step 1: Analyze cosine difference formula

The problem involves obtaining the cosine of the difference of two angles. The identity for this is \(\cos (A - B) = \cos A \cos B + \sin A \sin B\). Here, \(A = \sin^{-1} x, B = \cos^{-1} y\)

Step 2: Derive expressions using inverse trig functions

For \(A = \sin^{-1} x\), \(\cos A = \sqrt{1 - x^2}\) and \(\sin A = x\). For \(B = \cos^{-1} y\), \(\cos B = y\) and \(\sin B = \sqrt{1 - y^2}\)

Step 3: Apply to the cosine difference formula

Substitute these relations in the formula from step 1: \(\cos (A - B) = \cos A \cos B + \sin A \sin B = \sqrt{1 - x^2} * y + x * \sqrt{1 - y^2}\)

Key Concepts

Cosine Difference Formula

The cosine difference formula is a fundamental identity in trigonometry that helps us calculate the cosine of the difference between two angles. This formula is written as:\[ \cos(A - B) = \cos A \cos B + \sin A \sin B \]This identity is especially useful when dealing with trigonometric expressions involving inverse trigonometric functions. It allows us to transform complex expressions into simpler, algebraic forms that are easier to work with.
When you encounter a problem requiring the use of this formula, take note of the individual angles, often denoted as\( A \) and \( B \), and their trigonometric values. The formula leverages these values to express the final outcome without any trigonometric functions.

Algebraic Expression

Transforming trigonometric expressions into algebraic forms involves using inverse trigonometric functions and known identities. In the given exercise, we're finding an algebraic equivalent for:\[ \cos \left( \sin^{-1} x - \cos^{-1} y \right) \]We start by defining \( A = \sin^{-1} x \) and \( B = \cos^{-1} y \). From this:

• \( \cos A = \sqrt{1 - x^2} \)
• \( \sin A = x \)
• \( \cos B = y \)
• \( \sin B = \sqrt{1 - y^2} \)

By substituting these expressions into the cosine difference formula, we obtain:\[ \cos(A - B) = \sqrt{1 - x^2} \times y + x \times \sqrt{1 - y^2} \]This step allows us to move from a trigonometric representation typically involving angles to a more manageable algebraic format.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values within the relevant domain of the trigonometric functions included. They are powerful tools for simplifying expressions and solving equations in trigonometry. The exercise utilizes a few key identities:
Inverse trigonometric functions allow us to work backwards from ratios to angles. This is evident when analyzing expressions like \( \sin^{-1} x \) and \( \cos^{-1} y \). These angles are implicit in trigonometric identities, making them invaluable tools for transforming expressions.
Using fundamental identities, such as:

• Pythagorean identities \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Inverse transforms such as \( \sin^{-1} x\) giving \( \cos \theta = \sqrt{1-x^2} \)

These allow us to express \( \cos A \) and \( \sin B \) in terms of \( x \) and \( y \), ultimately leading to an easier form to comprehend and calculate.