Problem 15 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

(a) \(\frac{\sqrt{5}}{2}\), (b) \(\frac{\sqrt{34}}{5}\), (c) \(\frac{4\sqrt{15}}{15}\).

Step 1: Understanding Cotangent of Inverse Sine

The expression given is \( \cot \left(\sin^{-1} \frac{2}{3}\right) \). We denote \( \theta = \sin^{-1} \frac{2}{3} \). This means \( \sin \theta = \frac{2}{3} \). We need to find \( \cot \theta \), which is \( \frac{\cos \theta}{\sin \theta} \).

Step 2: Find \( \cos \theta \)

Using the Pythagorean identity, \( \cos \theta = \sqrt{1 - \sin^2 \theta} \). Therefore, \( \cos \theta = \sqrt{1 - \left(\frac{2}{3}\right)^2} = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3} \).

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

Now, compute \( \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\frac{\sqrt{5}}{3}}{\frac{2}{3}} = \frac{\sqrt{5}}{2} \).

Step 4: Understanding Secant of Inverse Tangent

The expression is \( \sec \left(\tan^{-1} -\frac{3}{5}\right) \). Let \( \phi = \tan^{-1} -\frac{3}{5} \). This means \( \tan \phi = -\frac{3}{5} \). We need to find \( \sec \phi = \frac{1}{\cos \phi} \).

Step 5: Find \( \cos \phi \) Using Pythagorean Identity

If \( \tan \phi = \frac{\text{opposite}}{\text{adjacent}} = -\frac{3}{5} \), use \( r = \sqrt{(-3)^2 + 5^2} = \sqrt{34} \) to find \( \cos \phi = \frac{5}{\sqrt{34}} \).

Step 6: Calculate \( \sec \phi \)

Thus, \( \sec \phi = \frac{1}{\cos \phi} = \frac{\sqrt{34}}{5} \).

Step 7: Understanding Cosecant of Inverse Cosine

The expression is \( \csc \left(\cos^{-1} -\frac{1}{4}\right) \). Let \( \gamma = \cos^{-1} -\frac{1}{4} \). This means \( \cos \gamma = -\frac{1}{4} \). We want to find \( \csc \gamma = \frac{1}{\sin \gamma} \).

Step 8: Find \( \sin \gamma \) Using Pythagorean Identity

\( \sin \gamma = \sqrt{1 - \cos^2 \gamma} = \sqrt{1 - \left(-\frac{1}{4}\right)^2} = \sqrt{\frac{15}{16}} = \frac{\sqrt{15}}{4} \).

Step 9: Calculate \( \csc \gamma \)

Finally, \( \csc \gamma = \frac{1}{\sin \gamma} = \frac{4}{\sqrt{15}} \). To rationalize, \( \csc \gamma = \frac{4\sqrt{15}}{15} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find angles when the values of trigonometric functions like sine, cosine, or tangent are known. These functions include \( ext{sin}^{-1}(x)\), \( ext{cos}^{-1}(x)\), and \( ext{tan}^{-1}(x)\), among others. They are the inverse operations of the standard trigonometric functions. When we write something like \(\sin^{-1}(\frac{2}{3})\), we are asking what angle \(\theta\) satisfies \(\sin \theta = \frac{2}{3}\).

Understanding these inverse functions is crucial since they help solve for angles in right triangles, especially when angles need to be expressed in radians or degrees.

• For \(\sin^{-1}(x)\), the range is \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
• For \(\cos^{-1}(x)\), the range is \([0, \pi]\).
• For \(\tan^{-1}(x)\), the range is \((-\frac{\pi}{2}, \frac{\pi}{2})\).

This means these functions are only defined when their output falls within these ranges, ensuring our angle solutions are correct and applicable to specific quadrants of the unit circle.

Pythagorean Identity

The Pythagorean identity is a fundamental relationship between the sides of a right triangle expressed through trigonometric functions. The primary identity is \(\sin^2 \theta + \cos^2 \theta = 1\).

This identity is especially useful when we know one trigonometric function value, like \(\sin \theta\), and need to find another, such as \(\cos \theta\).

• For example, if \(\sin \theta = \frac{2}{3}\), we can find \(\cos \theta\) by rearranging the identity: \(\cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - (\frac{2}{3})^2}\).

This elimination method helps us easily transition between different trigonometric concepts to find unknown sides or angles in the context of solving problems involving inverse trigonometric expressions. It is also useful for verifying the correctness of formulated expressions.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the trigonometric variables. They are foundational tools in trigonometry that simplify complex trigonometric expressions. Key identities include:

• Pythagorean Identities: Like mentioned, \(\sin^2 \theta + \cos^2 \theta = 1\) and its variations for \(\tan\) and \(\sec\).
• Reciprocal Identities: These involve interchanging main trigonometric functions with their reciprocals (e.g., \(\csc \theta = \frac{1}{\sin \theta}\), \(\sec \theta = \frac{1}{\cos \theta}\), \(\cot \theta = \frac{1}{\tan \theta}\)).
• Quotient Identities: Relate tangent and cotangent to sine and cosine (e.g., \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)).

Using these identities aids in solving complex trigonometric problems and simplifying expressions. They are immensely helpful in many areas of mathematics, including calculus and algebra. In our context, we make use of these identities to convert expressions such as \( \cot(\sin^{-1}(\frac{2}{3})) \) into their cotangent or reciprocal forms. This helps us find exact values efficiently.