Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-22: Find the exact value of the expression whenever it is defined. (a) \(\sin \left[\sin ^{-1} \frac{5}{13}-\cos ^{-1}\left(-\frac{3}{5}\right)\right]\) (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) \(-\frac{63}{65}\); (b) 0; (c) undefined

Step 1: Determine Individual Inverse Trigonometric Values for (a)

First, we solve for the angles from the given inverse trigonometric functions. We have: \( \sin^{-1} \frac{5}{13} \), which implies an angle \( \theta_1 \) such that \( \sin \theta_1 = \frac{5}{13} \). Similarly, \( \cos^{-1} \left(-\frac{3}{5}\right) \) implies an angle \( \theta_2 \) such that \( \cos \theta_2 = -\frac{3}{5} \).

Step 2: Find the Expression Inside the Sine Function for (a)

We need to compute \( \sin(\theta_1 - \theta_2) \). Notice that the corresponding angles are in the range determined by the inverse functions. \( \sin(\theta_1 - \theta_2) \) can be expressed as \( \sin \theta_1 \cos \theta_2 - \cos \theta_1 \sin \theta_2 \).

Step 3: Use Trigonometric Identities for (a)

Using the identity \( \sin \theta_1 = \frac{5}{13} \) and \( \cos \theta_2 = -\frac{3}{5} \), we find \( \cos \theta_1 = \frac{12}{13} \) (since \( 13^2 = 5^2 + 12^2 \)) and \( \sin \theta_2 = \frac{4}{5} \) (since \( 5^2 = 3^2 + 4^2 \)). Thus, \( \sin(\theta_1 - \theta_2) = \frac{5}{13} \times -\frac{3}{5} - \frac{12}{13} \times \frac{4}{5} = -\frac{63}{65} \).

Step 4: Calculate the Expression for (b)

For part (b), we want \( \cos(\sin^{-1} \frac{4}{5} + \tan^{-1} \frac{3}{4}) \). Let \( \alpha = \sin^{-1} \frac{4}{5} \) and \( \beta = \tan^{-1} \frac{3}{4} \). Then \( \cos \alpha = \frac{3}{5} \) and \( \sin \beta = \frac{3}{5} \), \( \cos \beta = \frac{4}{5} \). Use \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta = \frac{3}{5} \times \frac{4}{5} - \frac{4}{5} \times \frac{3}{5} = 0 \).

Step 5: Evaluate the Expression for (c)

For part (c), find \( \tan(\cos^{-1} \frac{1}{2} - \sin^{-1}(-\frac{1}{2})) \). \( \gamma = \cos^{-1} \frac{1}{2} \) (\( \gamma = \frac{\pi}{3} \)), \( \delta = \sin^{-1}(-\frac{1}{2}) \) (\( \delta = -\frac{\pi}{6} \)). Thus, \( \tan(\gamma - \delta) = \tan(\frac{\pi}{3} + \frac{\pi}{6}) = \tan(\frac{\pi}{2}) \), which is undefined.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in solving expressions involving trigonometric functions. They allow us to simplify complex expressions into more manageable forms. Some of the most important identities include:

• Sum and difference identities, such as \( \sin(a \pm b) = \sin a \cos b \pm \cos a \sin b \) and \( \cos(a \pm b) = \cos a \cos b \mp \sin a \sin b \).
• Pythagorean identities, like \( \sin^2 a + \cos^2 a = 1 \).

These identities are crucial in trigonometric calculus and geometry as they allow us to deduce unknown values. In Example (a), the sum and difference identities help express \( \sin(\theta_1 - \theta_2) \) in terms of known values of \( \sin \) and \( \cos \) for angles \( \theta_1 \) and \( \theta_2 \). Leveraging the identity, it becomes possible to calculate the exact trigonometric value from the inverse functions provided.

Angle Calculation

Calculating angles from inverse trigonometric functions involves deducing the angle whose trigonometric ratio is known. In Example (a), using \( \sin^{-1} \frac{5}{13} \) implies finding an angle \( \theta_1 \) for which \( \sin \theta_1 = \frac{5}{13} \). Similarly, \( \cos^{-1} (-\frac{3}{5}) \) answers the question: what angle \( \theta_2 \) has \( \cos \theta_2 = -\frac{3}{5} \)?Through drawing a right triangle or using the appropriate trigonometric theorem, you calculate complementary sides or angles. Remember, inverse functions give angles in specific ranges due to their periodic nature;

• \( \sin^{-1} \) gives angles between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).
• \( \cos^{-1} \) gives angles between 0 and \(\pi\).

This knowledge ensures accurate angle calculation, adding precision to your trigonometric solutions.

Exact Values

Finding exact values in trigonometry, particularly when working with inverse functions, revolves around understanding specific angle values and their corresponding trigonometric outcomes. Known trigonometric ratios for angles like \(30^\circ\), \(45^\circ\), and \(60^\circ\) help deduce exact values without resorting to a calculator.In Example (b), decomposing \( \sin^{-1} \frac{4}{5} \) gives a specific angle whose sine is \( \frac{4}{5} \), and \( \tan^{-1} \frac{3}{4} \) provides another angle with a tangent of \( \frac{3}{4} \). By substituting these values into trigonometric identities like \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \), you can derive an exact value \( (0) \) without using decimal approximations.Understanding and employing these methods not only simplifies trigonometric problems but also strengthens analytical and problem-solving skills by reducing reliance on numerical computations.