Problem 28 - Precalculus Mathematics for Calculus

Question

Find the exact value of the expression. $$\tan \left(\sin ^{-1} \frac{4}{5}\right)$$

Step-by-Step Solution

Answer

The exact value of the expression is \(\frac{4}{3}\).

Step 1: Identify the Inverse Function

Recognize that \(\sin^{-1} \left( \frac{4}{5}\right)\) represents an angle \(\theta\) such that \ \sin \theta = \frac{4}{5} \ and \(\theta\) is in the range \([-\pi/2, \pi/2]\).

Step 2: Use the Pythagorean Identity to Find \(\cos \theta\)

Since the angle \(\theta\) has sine value of \ \frac{4}{5} \, we use the Pythagorean identity: \ \sin^2 \theta + \cos^2 \theta = 1 \, to find \ \cos \theta \: \\[ \cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - \left(\frac{4}{5}\right)^2} = \sqrt{1 - \frac{16}{25}} = \sqrt{\frac{9}{25}} = \frac{3}{5}. \]

Step 3: Calculate \(\tan \theta\)

Now that we have both \(\sin \theta = \frac{4}{5} \) and \(\cos \theta = \frac{3}{5}\), calculate \(\tan \theta\). The tangent of \(\theta\) is the ratio of \ \sin \theta \ over \ \cos \theta\: \ \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3}. \]

Step 4: Conclude the Solution

Hence, the exact value of the expression \(\tan(\sin^{-1} \frac{4}{5})\) is \ \frac{4}{3} \.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to determine the angle whose trigonometric function takes a certain value. In the expression \( \sin^{-1} \left(\frac{4}{5}\right) \), the inverse sine function, denoted as \( \sin^{-1} \), is used to find an angle \( \theta \) such that \( \sin \theta = \frac{4}{5} \). This angle \( \theta \) falls within the range \([-\pi/2, \pi/2]\), which is the principal value range for inverse sine functions. In other words, the inverse sine function outputs angles that are restricted to this range, ensuring a unique solution. By utilizing inverse trigonometric functions, we can backtrack from a given sine, cosine, or tangent value to find their corresponding angle, which is crucial in applications like physics or engineering where angles need to be precisely calculated from trigonometric observations.

Pythagorean Identity

The Pythagorean identity is a fundamental equation in trigonometry that expresses the intrinsic relationship between the sine and cosine functions. It states: \( \sin^2 \theta + \cos^2 \theta = 1 \). In the context of our exercise, knowing that \( \sin \theta = \frac{4}{5} \), we utilize this identity to determine \( \cos \theta \). To find \( \cos \theta \), rearrange the identity to \( \cos \theta = \sqrt{1 - \sin^2 \theta} \). Substitute the known sine value: \[ \cos \theta = \sqrt{1 - \left(\frac{4}{5}\right)^2} = \sqrt{1 - \frac{16}{25}} = \sqrt{\frac{9}{25}} = \frac{3}{5}. \]The Pythagorean identity is not only helpful in solving trigonometric problems but also serves as a foundational concept that is widely used in deriving further trigonometric identities and solving complex equations.

Exact Values in Trigonometry

In trigonometry, it's essential to understand the exact values of trigonometric functions for specific angles, often requiring algebraic approaches rather than numerical approximations to compute accurately. The exercise highlights finding the exact value of \( \tan(\sin^{-1} \frac{4}{5}) \). Once we know both \( \sin \theta \) and \( \cos \theta \), computing the tangent, \( \tan \theta \), becomes straightforward.The formula for tangent is the ratio of sine to cosine:\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3}. \]Utilizing these exact values ensures precision in mathematical computations, especially important in areas requiring high accuracy like geometry or applied sciences. By understanding these trigonometric functions and their relationships, students can solve such problems confidently and without reliance on calculators, thereby developing a deeper comprehension of trigonometry.