Problem 43 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

\(\cos(A - B) = \frac{\sqrt{1-x^2} + xy}{\sqrt{1+y^2}}\).

Step 1: Set up the identity

We will use the identity for cosine of a difference: \[ \cos(A - B) = \cos A \cdot \cos B + \sin A \cdot \sin B \]where \(A = \sin^{-1} x\) and \(B = \tan^{-1} y\).

Step 2: Find \(\cos A\) and \(\sin A\)

Since \(A = \sin^{-1} x\), we know that \(\sin A = x\). To find \(\cos A\), we apply the Pythagorean identity:\[ \cos A = \sqrt{1 - \sin^2 A} = \sqrt{1 - x^2} \].

Step 3: Find \(\cos B\) and \(\sin B\)

Since \(B = \tan^{-1} y\), we know that \(\tan B = y\). In a right triangle, \( \tan B = \frac{\text{opposite}}{\text{adjacent}} \), which gives a ratio \( \frac{y}{1} \). This means the hypotenuse is \(\sqrt{1 + y^2}\). Thus, \(\cos B = \frac{1}{\sqrt{1 + y^2}}\) and \(\sin B = \frac{y}{\sqrt{1 + y^2}}\).

Step 4: Substitute into the cosine difference formula

Substitute the values from Steps 2 and 3 into the formula from Step 1:\[ \cos(A - B) = \sqrt{1 - x^2} \cdot \frac{1}{\sqrt{1 + y^2}} + x \cdot \frac{y}{\sqrt{1 + y^2}} \].

Step 5: Simplify the expression

Combine the terms under a common denominator:\[ \cos(A - B) = \frac{\sqrt{1 - x^2}}{\sqrt{1 + y^2}} + \frac{xy}{\sqrt{1 + y^2}} = \frac{\sqrt{1 - x^2} + xy}{\sqrt{1 + y^2}} \].

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are very interesting because they allow us to work with angles when the trigonometric value is known. Common inverse trigonometric functions include:

• \( \sin^{-1} \) (also known as arcsin)
• \( \cos^{-1} \) (also known as arccos)
• \( \tan^{-1} \) (also known as arctan)

These functions tell us what angle corresponds to a particular sine, cosine, or tangent value. When you read \( \sin^{-1} x \), it translates to "what angle has a sine of \( x \)?"
This is hugely useful in calculus and geometry.In our problem, \( A = \sin^{-1} x \), which means if \( \sin A = x \), we are using these functions to calculate exact values for angles and find related trigonometric values related to those angles, such as cosine in this exercise.

Cosine Difference Formula

The cosine difference formula is very useful for finding the cosine of an angle that is the difference between two other angles. The formula is: \[ \cos(A - B) = \cos A \cdot \cos B + \sin A \cdot \sin B \]This formula helps break down complex angle expressions into simpler parts that can be managed more easily. You can think of it as a handy tool that enables you to work with known values instead of trying to calculate cosine directly for awkward angles.By applying this formula in the exercise, the problem becomes more manageable. We break the appearance of \( \cos(\sin^{-1} x - \tan^{-1} y) \) into simpler cosine and sine pairs: \( \cos A \) and \( \sin A \), \( \cos B \) and \( \sin B \).
This strategic breakdown makes it easier to substitute known values and simplify calculations.

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry that connects the squares of the sine and cosine of an angle. It is fundamentally expressed as: \[ \sin^2 A + \cos^2 A = 1 \]This identity is derived from the Pythagorean theorem related to right triangles, and it is exceptionally useful when finding unknown trigonometric values.For \( A = \sin^{-1} x \) in this exercise, the identity allows us to calculate \( \cos A \) by rearranging the identity to: \[ \cos A = \sqrt{1 - \sin^2 A} \] Since \( \sin A = x \), it simplifies to \( \cos A = \sqrt{1 - x^2} \).
With this, we get a vital component of our trigonometric expression needed for further calculations in the exercise.

Angle Subtraction

Angle subtraction in trigonometry involves working out the trigonometric functions of differences in angles. Trigonometric identities like the cosine difference formula simplify these equations.In our exercise, we're specifically interested in subtracting two angle measures: \( A = \sin^{-1} x \) and \( B = \tan^{-1} y \). By understanding each angle measure and applying the cosine difference formula, we gain a deeper understanding of how angle subtraction works.
This results in breaking down the complex trigonometric expression into simpler components that have known trigonometric values.Using angle subtraction principles, we handle complex trigonometric manipulations more competently.

• Apply identity formulas effectively, like the cosine difference formula
• Reduce complexity by substituting known values at each step
• Make calculations less error-prone by related breakdowns

Understanding angle subtraction greatly enhances problem-solving skills in trigonometry.