Problem 44 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding Inverse Trigonometric Identities

Let's start by recalling the identities for inverse trigonometric functions. We know that for any angle \( heta \), \( \sin^{-1}(x) \) gives an angle whose sine is \( x \), and \( \cos^{-1}(y) \) gives an angle whose cosine is \( y \).

Step 2: Applying Inverse Sine Identity

Let \( \theta_1 = \sin^{-1}(x) \), which implies \( \sin(\theta_1) = x \). Then we can deduce \( \cos(\theta_1) = \sqrt{1-x^2} \) because \( \sin^2(\theta_1) + \cos^2(\theta_1) = 1 \).

Step 3: Applying Inverse Cosine Identity

Let \( \theta_2 = \cos^{-1}(y) \), which implies \( \cos(\theta_2) = y \). Therefore, we deduce \( \sin(\theta_2) = \sqrt{1-y^2} \) as \( \sin^2(\theta_2) + \cos^2(\theta_2) = 1 \).

Step 4: Using the Angle Sum Identity for Tangent

We want to find \( \tan(\theta_1 + \theta_2) \). The angle sum formula for tangent is \( \tan(\theta_1 + \theta_2) = \frac{\tan(\theta_1) + \tan(\theta_2)}{1 - \tan(\theta_1) \tan(\theta_2)} \).

Step 5: Finding Tangent Values for Each Angle

We have \( \tan(\theta_1) = \frac{\sin(\theta_1)}{\cos(\theta_1)} = \frac{x}{\sqrt{1-x^2}} \) and \( \tan(\theta_2) = \frac{\sin(\theta_2)}{\cos(\theta_2)} = \frac{\sqrt{1-y^2}}{y} \).

Step 6: Substituting into the Tangent Sum Formula

Substitute \( \tan(\theta_1) \) and \( \tan(\theta_2) \) into the angle sum formula:\[\tan(\theta_1 + \theta_2) = \frac{ \frac{x}{\sqrt{1-x^2}} + \frac{\sqrt{1-y^2}}{y} }{ 1 - \left( \frac{x}{\sqrt{1-x^2}} \right) \left( \frac{\sqrt{1-y^2}}{y} \right) }\].

Step 7: Simplifying the Expression

Simplify the numerator and the denominator of the expression obtained:\[\text{Numerator: } \frac{xy + y\sqrt{1-y^2}}{y}\sqrt{1-x^2}\]\[\text{Denominator: } 1 - \frac{x\sqrt{1-y^2}}{y\sqrt{1-x^2}}\]Both terms can be simplified further to obtain a form purely in terms of \( x \) and \( y \).

Step 8: Final Simplification (if needed)

Ensure that the final expression can be reduced or simplified further for ease of interpretation. If required, multiply through by common denominators to clear any fractions.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential when you want to find the angle associated with a given trigonometric ratio. For example, the function \( \sin^{-1}(x) \) represents the angle whose sine value is \( x \). Similarly, \( \cos^{-1}(y) \) indicates the angle whose cosine is \( y \). These functions are the inverse operations of regular trigonometric functions. They allow us to work backwards from angles to ratios.

One important point about these functions is their range. The principal value of \( \sin^{-1}(x) \) is between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), while \( \cos^{-1}(y) \) falls between \(0\) and \(\pi\). Understanding these ranges helps in accurately calculating angles and utilizing inverse trigonometric functions in various problems. When substituting these inverse functions into an expression, converting them to simple sine and cosine values can make your life easier.

Through the step-by-step process described in the exercise, you would identify key angles by labeling \( \theta_1 = \sin^{-1}(x) \) and \( \theta_2 = \cos^{-1}(y) \). This gives us working angles to apply further trigonometric identities necessary to solve the problem.

Angle Sum Identity

The angle sum identity is a powerful tool in trigonometry that helps calculate the trigonometric function of a sum of two angles using the functions of the angles themselves. For instance, the tangent angle sum identity states that:\[\tan(\theta_1 + \theta_2) = \frac{\tan(\theta_1) + \tan(\theta_2)}{1 - \tan(\theta_1)\tan(\theta_2)}\]This identity allows us to find the tangent of the sum of two angles if we know the tangent of each angle.

In our exercise, we want to find \( \tan(\sin^{-1}(x) + \cos^{-1}(y)) \). By expressing these angles individually in terms of their respective sines and cosines, you can plug them into the angle sum identity formula for tangent. This creates a bridge from individual inverse functions to a combined trigonometric expression.

When using this identity, ensure each component, like \( \tan(\theta_1) \) and \( \tan(\theta_2) \), is expressed in terms of the known values, \( x \) and \( y \). This makes solving the expression much more manageable.

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the angles involved and are vital for simplifying complicated expressions. The basic ones you'll frequently encounter include Pythagorean identities like \( \sin^2(\theta) + \cos^2(\theta) = 1 \). This identity is pivotal in converting between sine and cosine, allowing you to express one in terms of the other.

In this exercise, both \( \theta_1 \) and \( \theta_2 \) used this fundamental identity. For \( \theta_1 = \sin^{-1}(x) \), applying the identity gave \( \cos(\theta_1) = \sqrt{1-x^2} \). Similarly, for \( \theta_2 = \cos^{-1}(y) \), it derived \( \sin(\theta_2) = \sqrt{1-y^2} \). These conversions are essential for progressing through many types of trigonometric proofs and calculations.

By employing these identities, you avoid complex computations and can convert your target expression, such as \( \tan(\theta_1 + \theta_2) \), into its simplest form comprising only the variables \( x \) and \( y \). These identities serve as shortcuts that make calculations significantly simpler.

Simplifying Trigonometric Expressions

Simplification is the process of transforming a complicated expression into a more manageable and readable form. With trigonometric expressions, this often involves using identities, rationalizing denominators, and clearing up fractions. The goal is to rewrite complex formulas in a simplified form involving only some basic components like \( x \) and \( y \).

For this exercise, once \( \tan(\sin^{-1}(x) + \cos^{-1}(y)) \) was set up using the angle sum formula, each part needed simplification. Both the numerator and the denominator were initially constructed using the derived values of \( \tan(\theta_1) \) and \( \tan(\theta_2) \). From there, the fractions \( \frac{x}{\sqrt{1-x^2}} \) and \( \frac{\sqrt{1-y^2}}{y} \) were combined and simplified.

The simplification step is crucial because it allows for practical application in further calculations or proofs. Here, you multiply through by common denominators, aiming to eliminate complex fractions and thereby arrive at a clear expression that's easy to interpret and use in subsequent math problems.