Problem 25 - Thomas Calculus

Question

Find the values in Exercises \(17-28\) $$ \sec \left(\tan ^{-1} 1+\csc ^{-1} 1\right) $$

Step-by-Step Solution

Answer

The value is \(-\sqrt{2}\).

Step 1: Understand Inverse Trigonometric Functions

Recognize that \(\tan^{-1}(1)\) is asking for an angle whose tangent is 1 and \(\csc^{-1}(1)\) is asking for an angle whose cosecant is 1.

Step 2: Solve for \( \tan^{-1}(1) \)

Since \( \tan(\frac{\pi}{4}) = 1 \), we know that \( \tan^{-1}(1) = \frac{\pi}{4} \). This is because \(\frac{\pi}{4}\) radians (or 45 degrees) is the angle whose tangent is equal to 1.

Step 3: Solve for \( \csc^{-1}(1) \)

Recalling that \( \csc(\theta) = \frac{1}{\sin(\theta)} \), if \( \csc(\theta) = 1 \), then \( \sin(\theta) = 1 \). The angle \( \theta \) where this occurs is \( \frac{\pi}{2} \). Hence, \( \csc^{-1}(1) = \frac{\pi}{2} \).

Step 4: Calculate the Sum of the Angles

Add the two angles obtained: \( \tan^{-1}(1) + \csc^{-1}(1) \). This gives \( \frac{\pi}{4} + \frac{\pi}{2} = \frac{3\pi}{4} \).

Step 5: Evaluate \( \sec(\frac{3\pi}{4}) \)

Recall that \( \sec(\theta) = \frac{1}{\cos(\theta)} \). At \( \theta = \frac{3\pi}{4} \), we have \( \cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2} \), so \( \sec(\frac{3\pi}{4}) = -\sqrt{2} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools in mathematics to determine angles when given a trigonometric function value. They effectively reverse the role of ordinary trigonometric functions, allowing us to work backward from a ratio to an angle. For instance, when we see \( \tan^{-1}(1) \), we understand it asks for the angle whose tangent is 1.

Inverse functions include:

• \( \sin^{-1}(x) \) — for finding angles with a given sine
• \( \cos^{-1}(x) \) — for finding angles using cosine
• \( \tan^{-1}(x) \) — to determine angles from tangent values
• Similarly, for secant, cosecant, and cotangent

Each has its own specific range, ensuring the output is a valid angle. Leveraging these inverse functions is key to solving the problem, converting values back into angles.

Angle Sum Identities

In trigonometry, angle sum identities allow us to find the sine, cosine, and tangent of a sum of two angles. These identities are pivotal when dealing with expressions such as \( \tan^{-1}(1) + \csc^{-1}(1) \).
For sums, the relevant identities include:

• \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)
• \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)
• \( \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \) when both \( \tan a \) and \( \tan b \) are defined

These identities are more than just formulas; they help simplify complex expressions combining multiple angles. While the problem does not directly require using angle sum identities, understanding them enriches comprehension of related trigonometric summations.

Secant Function

The secant function is a crucial trigonometric function defined as the reciprocal of the cosine of an angle, written mathematically as \( \sec(\theta) = \frac{1}{\cos(\theta)} \). This function helps evaluate the trigonometric ratio at a given angle, even when cosine is negative or zero isn't directly possible.
For solving \( \sec(\frac{3\pi}{4}) \), we need to first determine \( \cos(\frac{3\pi}{4}) \). Knowing that \( \cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2} \), calculating the secant is straightforward:

• Find the reciprocal: \( \sec(\frac{3\pi}{4}) = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2} \)

In essence, the secant function allows us to explore further trigonometric properties, ensuring comprehensive solutions for trigonometric expressions.