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Chapter 3: Problem 10

Calculate the value of the given inverse trigonometric function at the given point. $$ \operatorname{arcsec}(2) $$

Short Answer

Expert verified
\(\operatorname{arcsec}(2) = \frac{\pi}{3}\)

Step by step solution

01

Understanding the arcsec function

The arcsecant function, denoted as \(\operatorname{arcsec}(x)\), is the inverse of the secant function. This function takes a number \(x\) and returns an angle \(\theta\) such that \(\operatorname{sec}(\theta) = x\). The value range of \(\theta\) is usually \([0, \pi/2) \cup (\pi/2, \pi]\).
02

Determine the secant value needed

To find \(\operatorname{arcsec}(2)\), we need to determine an angle \(\theta\) such that \(\operatorname{sec}(\theta) = 2\). Recall that \(\operatorname{sec}(\theta) = \frac{1}{\cos(\theta)}\), which implies \(\cos(\theta) = \frac{1}{2}\).
03

Find the angle using the cosine value

We need to find an angle \(\theta\) such that \(\cos(\theta) = \frac{1}{2}\). The angle \(\theta\) satisfying \(\cos(\theta) = \frac{1}{2}\) within the specified range \([0, \pi/2) \cup (\pi/2, \pi]\) is \(\theta = \frac{\pi}{3}\).
04

Conclusion

Therefore, the value of \(\operatorname{arcsec}(2)\) is \(\frac{\pi}{3}\) because \(\operatorname{sec}(\theta) = 2\) corresponds to \(\cos(\theta) = \frac{1}{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

arcsecant
The arcsecant function, often written as \(\operatorname{arcsec}(x)\), is the inverse of the secant function. It helps us find angles associated with a specific secant value. Here's how it works:
  • The function \(\operatorname{arcsec}(x)\) returns an angle \(\theta\) such that the secant of \(\theta\) equals \(x\).
  • Across different calculators and textbooks, the range for the angle \(\theta\) is typically \([0, \pi/2) \cup (\pi/2, \pi]\).
  • This range avoids the values where the secant function is undefined, ensuring that the arcsecant function remains valid and accurate.
Understanding this function is key to solving equations involving trigonometric inverses, as it links the secant value directly to its corresponding angle.
secant function
The secant function is one of the fundamental trigonometric functions. It's closely related to the cosine function. When people talk about the secant of an angle \(\theta\):
  • They are referring to the reciprocal of the cosine function. This means \(\operatorname{sec}(\theta) = \frac{1}{\cos(\theta)}\).
  • The secant function is undefined wherever cosine equals zero because division by zero is not possible.
  • In terms of a circle, the secant function represents the length of the hypotenuse divided by the adjacent side in a right triangle.
Since \(\operatorname{sec}(\theta)\) is the reciprocal of \(\cos(\theta)\), it naturally takes larger and positive values, especially above 1 or less than -1, increasing towards infinity as \(\cos(\theta)\) approaches zero.
angle calculation
Calculating angles using trigonometric values often requires an understanding of inverse functions. Here鈥檚 how one calculates angles from trigonometric values:
  • To find the angle \(\theta\) for a given trigonometric value, like when \(\cos(\theta) = \frac{1}{2}\), using inverse functions can directly point you to the solutions.
  • For example, if \(\operatorname{sec}(\theta) = 2\), it translates to finding \(\theta\) where \(\cos(\theta) = \frac{1}{2}\).
  • Within the range of the arcsecant function, this cosine result correctly identifies the angle \(\theta = \frac{\pi}{3}\).
These calculations demand familiarity with the specific ranges of each inverse function to ensure the angle falls within acceptable limits.
trigonometric identities
Trigonometric identities are equations that involve trigonometric functions and are universally true for all angles. They provide essential tools for simplifying and solving trigonometric expressions:
  • One fundamental identity is \(\operatorname{sec}(\theta) = \frac{1}{\cos(\theta)}\), connecting secant and cosine functions.
  • These identities are useful for transforming complex trigonometric expressions into simpler, more manageable forms.
  • They also serve as the foundation for evaluating inverse functions like \(\operatorname{arcsec}(x)\) by linking them to known angles and trigonometric results.
Being conversant with these identities helps you navigate through problems involving multiple trigonometric functions.

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Most popular questions from this chapter

Calculate the value of the given inverse trigonometric function at the given point. $$ \operatorname{arccsc}(-\sqrt{2}) $$

Differentiate the given expression with respect to \(x\). $$ \tanh ^{-1}(\operatorname{coth}(x)) $$

\- A real-valued function \(f\) of a real variable \(x\) is said to be algebraic if there is a polynomial \(p(u, v)\) with integer coefficients such that \(p(x, f(x))=0\) for all \(x .\) For example, \(f(x)=\) \(2 x+\sqrt{x^{2}+1}\) is algebraic because $$ \begin{aligned} p(x, f(x))=& 3 x^{2}+\left(2 x+\sqrt{x^{2}+1}\right)^{2} \\ &-4 x\left(2 x+\sqrt{x^{2}+1}\right)-1 \\ \equiv & 0 \end{aligned} $$ for \(p(u, v)=3 u^{2}+v^{2}-4 u v-1 .\) A function that is not algebraic is said to be transcendental. Find a polynomial that shows that the given expression is algebraic. $$ x-\sqrt{2} $$

A function \(f,\) a point \(c,\) an increment \(\Delta x,\) and a positive integer \(n\) are given. Use the method of increments to estimate \(f(c+\Delta x)\). Then let \(h=\Delta x\) / \(N\). Use the method of increments to obtain an estimate \(y_{1}\) of \(f(c+h) .\) Now, with \(c+h\) as the base point and \(y_{1}\) as the value of \(f(c+h),\) use the method of increments to obtain an estimate \(y_{2}\) of \(f(c+2 h)\). Continue this process until you obtain an estimate \(y_{N}\) of \(f(c+N \cdot h)=f(c+\Delta x) .\) We say that we have taken \(N\) steps to obtain the approximation. The number \(h\) is said to be the step size. Use a calculator or computer to evaluate \(f(c+\Delta x)\) directly. Compare the accuracy of the one step and \(N\) -step approximations. $$ f(x)=\sqrt{x}, c=4, \Delta x=0.5, N=5 $$

Calculate the value of the given inverse trigonometric function at the given point. $$ \arctan (\tan (-3 \pi / 4)) $$
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