Problem 6 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-10, find the exact value without using a calculator. $$ \operatorname{arcsec}(2) $$

Step-by-Step Solution

Answer

The exact value is \( \frac{\pi}{3} \).

Step 1: Understand the Definition of Arcsec

The arcsecant function, \( \operatorname{arcsec}(x) \), is the inverse of the secant function. It returns the angle \( \theta \) whose secant is \( x \). This means if \( \theta = \operatorname{arcsec}(2) \), then \( \sec(\theta) = 2 \).

Step 2: Recall the Secant Function Identity

The secant function is defined as the reciprocal of the cosine function. Therefore, \( \sec(\theta) = \frac{1}{\cos(\theta)} \). This means \( \frac{1}{\cos(\theta)} = 2 \).

Step 3: Solve for Cosine

From the equation \( \frac{1}{\cos(\theta)} = 2 \), solve for \( \cos(\theta) \). Multiplying both sides by \( \cos(\theta) \) and dividing both sides by 2 gives \( \cos(\theta) = \frac{1}{2} \).

Step 4: Determine the Angle for Cosine Value

The angle \( \theta \) for which \( \cos(\theta) = \frac{1}{2} \) in the typical range of arcsecant \([0, \pi] \), excluding \(\frac{\pi}{2}\), is \( \frac{\pi}{3} \). This is because \( \cos(\frac{\pi}{3}) = \frac{1}{2} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to determine the angles when given a trigonometric ratio. These functions are the reverse process of the trigonometric functions themselves and they include:

• Arc Sine (\( \operatorname{arcsin}(x) \))
• Arc Cosine (\( \operatorname{arccos}(x) \))
• Arc Tangent (\( \operatorname{arctan}(x) \))
• Arc Cosecant (\( \operatorname{arccsc}(x) \))
• Arc Secant (\( \operatorname{arcsec}(x) \))
• Arc Cotangent (\( \operatorname{arccot}(x) \))

The arcsecant function, \( \operatorname{arcsec}(x) \), is particularly important for finding angles when given the secant value. It returns the angle \( \theta \) for which the secant of \( \theta \) is \( x \). For instance, if \( \operatorname{arcsec}(2) = \theta \), then \( \sec(\theta) = 2 \). The result of this function is typically an angle within the range \( [0, \pi] \), excluding \( \frac{\pi}{2} \), as secant is undefined there.

Secant Function

The secant function is a type of trigonometric function which is notable for being the reciprocal of the cosine function. It is defined as:\[ \sec(\theta) = \frac{1}{\cos(\theta)} \]This means that if you know \( \cos(\theta) \), you can find \( \sec(\theta) \) by taking the reciprocal of the cosine value. Consequently, when you see \( \sec(\theta) = 2 \), it implies \( \cos(\theta) = \frac{1}{2} \), because \( 2 = \frac{1}{\frac{1}{2}} \).

The secant function can sometimes be challenging to work with because it is undefined for values where the cosine function is zero (such as at \( \theta = \frac{\pi}{2} \)). This is because a reciprocal of zero is undefined. In practice, this makes the graph of \( \sec(\theta) \) appear as a series of upward and downward curves, punctuated by vertical asymptotes where it is undefined.

Cosine Function

The cosine function is one of the primary trigonometric functions and is denoted by \( \cos(\theta) \). It is essential in determining the relationship between a right triangle's angle and the lengths of its adjacent side and hypotenuse. For an angle \( \theta \), \( \cos(\theta) \) is given by:\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]Cosine values range from -1 to 1, and this function helps in various applications ranging from solving triangles to wave analysis in physics. In the context of finding \( \operatorname{arcsec}(2) \), understanding that \( \sec(\theta) \) being the reciprocal of \( \cos(\theta) \) means, when \( \sec(\theta) = 2 \), \( \cos(\theta) = \frac{1}{2} \). This specific cosine value of \( \frac{1}{2} \) is special, corresponding to the angle \( \theta = \frac{\pi}{3} \), because \( \cos(\frac{\pi}{3}) = \frac{1}{2} \). This understanding helps to determine the exact value of trigonometric functions analytically without relying on a calculator.