Problem 18 - Precalculus: Functions and Graphs

Question

Find the exact value of the expression whenever It is defined. (a) \(\sin \left[\sin ^{-1} \frac{s}{13}-\cos ^{-1}\left(-\frac{3}{5}\right)\right]\) (b) \(\cos \left(\sin ^{-1} \frac{4}{5}+\tan ^{-1} \frac{3}{4}\right)\) (c) \(\tan \left[\cos ^{-1} \frac{1}{2}-\sin ^{-1}\left(-\frac{1}{2}\right)\right]\)

Step-by-Step Solution

Answer

(a) Depends on s, (b) is 0, (c) approximately \( \frac{4}{\sqrt{3}} \).

Step 1: Understand the Inverse Trigonometric Functions

We start by recognizing the properties of inverse trigonometric functions \( \sin^{-1}(x) \), \( \cos^{-1}(x) \), and \( \tan^{-1}(x) \). They give us angles whose sine, cosine, and tangent are \( x \) respectively. This helps in recognizing the angles involved in each part of the expressions.

Step 2: Solve Part (a)

In part (a):1. \( \sin^{-1}(\frac{s}{13}) \) is the angle \( \theta \) such that \( \sin(\theta) = \frac{s}{13} \).2. \( \cos^{-1}(-\frac{3}{5}) \) is the angle \( \phi \) such that \( \cos(\phi) = -\frac{3}{5} \). So, \( \phi = \pi - \cos^{-1}(\frac{3}{5}) \).3. Use sine subtraction formula, \( \sin(\theta - \phi) = \sin \theta \cos \phi - \cos \theta \sin \phi \).4. Solve: Given \( \phi = \pi - \cos^{-1}(\frac{3}{5}) \), this implies that \( \cos(\phi) = \frac{3}{5} \) and \( \sin(\phi) = \pm \sqrt{1-(\frac{3}{5})^2} = \pm \frac{4}{5} \) and \( \phi \) lies in second quadrant (since \( \cos^{-1}(-\frac{3}{5}) \) provides an obtuse angle).5. Substituting values and simplifying terms, we get \( \sin \left( \sin^{-1} \frac{s}{13} - \cos^{-1} \left(- \frac{3}{5} \right) \right) = \frac{s}{13} \left(-\frac{3}{5}\right) - \sqrt{1-\left(\frac{s}{13}\right)^2} \times \frac{4}{5} \), assuming \( s \leq 13 \) (since only then it can be a defined angle in triangles).

Step 3: Solve Part (b)

In part (b):1. \( \sin^{-1}(\frac{4}{5}) \) is the angle \( \theta \) such that \( \sin(\theta) = \frac{4}{5} \).2. \( \tan^{-1}(\frac{3}{4}) \) is the angle \( \phi \) such that \( \tan(\phi) = \frac{3}{4} \).3. Use cosine addition formula, \( \cos(\theta + \phi) = \cos \theta \cos \phi - \sin \theta \sin \phi \).4. Since \( \sin(\theta) = \frac{4}{5}, \cos(\theta) = \frac{3}{5} \); \( \tan(\phi) = \frac{3}{4} \) yielding \( \cos(\phi) = \frac{4}{5} \) and \( \sin(\phi) = \frac{3}{5} \).5. Substitute these values into the formula: \( \cos(\sin^{-1} \frac{4}{5} + \tan^{-1} \frac{3}{4}) = \frac{3}{5} \frac{4}{5} - \frac{4}{5} \frac{3}{5} = 0 \).

Step 4: Solve Part (c)

In part (c):1. \( \cos^{-1}(\frac{1}{2}) \) is the angle \( \theta \) such that \( \cos(\theta) = \frac{1}{2} \), so \( \theta = \frac{\pi}{3} \).2. \( \sin^{-1}(-\frac{1}{2}) \) is the angle \( \phi \) such that \( \sin(\phi) = -\frac{1}{2} \), so \( \phi = -\frac{\pi}{6} \).3. Use tangent subtraction formula, \( \tan(\theta - \phi) = \frac{\tan \theta - \tan \phi}{1 + \tan \theta \tan \phi} \).4. Since, \( \tan(\theta) = \sqrt{3} \), \( \tan(\phi) = -\frac{1}{\sqrt{3}} \).5. Substitute these values into the formula: \( \tan(\cos^{-1} \frac{1}{2} - \sin^{-1}(-\frac{1}{2})) = \frac{\sqrt{3} + \frac{1}{\sqrt{3}}}{1 + \sqrt{3} \left(-\frac{1}{\sqrt{3}}\right)} = \frac{\frac{4}{\sqrt{3}}}{1} = \frac{4}{\sqrt{3}} \approx \frac{4}{\sqrt{3}} \).

Step 5: Verify and Conclude

Go through each derived equation and ensure the domain of the trigonometric functions and inverse functions hold true, ensuring the provided values for inverse trig functions match conditions \( |x| \leq 1 \) for sine and cosine. This confirms each part was correctly evaluated, leading to: (a) Dependent on the specific value of \( s \), (b) = 0, (c) Unable to find a clear simplification without a calculator, therefore approximate calculations suffice or a more evaluative approach might yield other insights.

Key Concepts

Sine Subtraction Formula

The sine subtraction formula is a trigonometric identity used to find the sine of the difference of two angles. It is given by the expression:

• \[\sin(\theta - \phi) = \sin \theta \cos \phi - \cos \theta \sin \phi\]

This formula helps us decompose complex trigonometric problems into simpler parts that involve sine and cosine of individual angles. By applying this, you can manage complex expressions involving inverse trigonometric functions.
The essence is to convert expressions with inverse sine and inverse cosine into an expression solely involving sine by considering angles derived from recognized trigonometric values. Part (a) of the exercise demonstrates this application, where initially you derive values for \(\phi\) and \(\theta\) using inverse functions, and then substitute into the sine subtraction formula. Associating these individual angles to their sine and cosine components facilitates the calculation down to basic arithmetic operations.

Cosine Addition Formula

The cosine addition formula is quite useful when dealing with the sum of two angles, derived as:

• \[\cos(\theta + \phi) = \cos \theta \cos \phi - \sin \theta \sin \phi\]

This particular identity enables simplification of expressions that add angles. In Part (b) of the exercise, this formula shines as it takes an expression that first captures individual angles using inverse sine and inverse tangent before combining them through cosine addition.
In this context, knowing that \( \sin^{-1}(\frac{4}{5}) \) and \( \tan^{-1}(\frac{3}{4}) \) give us specific angle values helps compute \(\cos \theta\) and \(\sin \theta\), and \(\cos \phi\) and \(\sin \phi\), respectively. The cosine addition formula hence becomes the crucial technique for breaking down the expression to arrive at the conclusion \(0\), evidencing how these trigonometric identities streamline computation.

Tangent Subtraction Formula

To simplify expressions having tangent with angles' differences, the tangent subtraction formula is applied. It is represented as:

• \[\tan(\theta - \phi) = \frac{\tan \theta - \tan \phi}{1 + \tan \theta \tan \phi}\]

This formula helps simplify the subtraction of angles into an expression that involves only basic operations of tangent values from each angle.
In Part (c) of the exercise, the angles are derived using \(\cos^{-1}\) and \(\sin^{-1}\) expressions for \(\theta\) and \(\phi\). Here, \(\tan \theta\) and \(\tan \phi\) are calculated individually before incorporating them into the tangent subtraction formula. Understanding the geometric relationships that define these angles provides an efficient tool for evaluating tangent of angle differences, although the computation concludes with a continued reliance on further evaluative or numerical methods for approximation when necessary, as indicated by the approximated outcome \(\frac{4}{\sqrt{3}}\).