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Chapter 10: Problem 6

Find the exact value or state that it is undefined. $$ \sec \left(-\frac{3 \pi}{2}\right) $$

Short Answer

Expert verified
\(\sec \left(-\frac{3 \pi}{2}\) is undefined because it involves division by zero.

Step by step solution

01

Determine the Reference Angle

The angle \(-\frac{3 \pi}{2}\) can be converted into a positive angle by adding \(2\pi\) until we fall into the range of a full rotation \([0, 2\pi)\). Since \(2\pi = \frac{4\pi}{2}\), we have \(-\frac{3\pi}{2} + 2\pi = \frac{\pi}{2}\). The reference angle is \(\frac{\pi}{2}\).
02

Understand the Position on the Unit Circle

In the unit circle, \(\frac{\pi}{2}\) corresponds to the point (0, 1) on the circle since it is at the top of the vertical axis. Here, the cosine of \(\frac{\pi}{2}\) is 0.
03

Recall the Definition of Secant

The secant function is defined as the reciprocal of the cosine function. Thus, \(\sec \theta = \frac{1}{\cos \theta}\). Here, \(\cos \frac{\pi}{2} = 0\).
04

Evaluate the Secant for the Angle

Since \(\cos \frac{\pi}{2} = 0\), the reciprocal, given by \(\sec \frac{\pi}{2} = \frac{1}{0}\), does not exist. Division by zero is undefined in mathematics.

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Key Concepts

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

Unit Circle
The Unit Circle is a fundamental concept in trigonometry. It helps us understand angles and their corresponding sine, cosine, and tangent values. The unit circle is a circle with a radius of 1 unit centered at the origin of a coordinate plane.

A key feature of the unit circle is that for any given angle, the x-coordinate of the point on the circle represents the cosine of that angle, while the y-coordinate represents the sine. These coordinates are crucial when working with trigonometric functions.

To find angles on the unit circle:
  • Angles are measured in radians, though degrees can also be used.
  • The full circle represents a complete rotation of either 360 degrees or \(2\pi\) radians.
  • Commonly used angles include \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\), and \(2\pi\), situated at important points in the unit circle.
Understanding the unit circle allows you to easily find the sine, cosine, and tangent of any angle, as well as their respective reciprocals.
Secant Function
The Secant Function is a trigonometric function related closely to the cosine function. It is defined as the reciprocal of the cosine function. Simply put, for any angle \(\theta\), the secant is given by:
  • \( \sec \theta = \frac{1}{\cos \theta} \).
This means that wherever the cosine function is zero, the secant function becomes undefined. This happens because division by zero is not allowed in mathematics, leading to undefined expressions.

To know the secant function better:
  • The secant values become larger as the cosine approaches zero from either side of the circle.
  • While \( \cos \theta \) ranges from -1 to 1, \( \sec \theta \) can take values from negative or positive infinity whenever cosine is zero.
  • The secant function is especially important when addressing problems involving vertical asymptotes, as it tends to infinity at those points.
Using the secant function provides a deeper understanding of angle relationships beyond the primary sine and cosine considerations.
Reference Angle
A Reference Angle is a crucial tool when dealing with angles in trigonometry. It is the acute version of an angle, always between 0 and \( \frac{\pi}{2} \), that shares the same terminal side in standard position.


How to calculate a reference angle:
  • If an angle is in the first quadrant, the reference angle is the angle itself.
  • In the second quadrant, subtract the angle from \( \pi \).
  • For the third quadrant, subtract \( \pi \) from the angle.
  • In the fourth quadrant, subtract the angle from \( 2\pi \).
By finding the reference angle, we simplify problems because trigonometric function values for any angle can be determined based on its reference angle. It's especially beneficial when the angle is given in a form that does not instantly reveal its position in the unit circle.

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Most popular questions from this chapter

In Exercises 222 - 233 , find the domain of the given function. Write your answers in interval notation. $$ f(x)=\operatorname{arcsec}(12 x) $$

In Exercises \(131-154\), find the exact value or state that it is undefined. $$ \csc \left(\arcsin \left(\frac{3}{5}\right)\right) $$

In Exercises \(165-184\), rewrite the quantity as algebraic expressions of \(x\) and state the domain on which the equivalence is valid. $$ \sin (2 \arcsin (7 x)) $$

In Exercises \(165-184\), rewrite the quantity as algebraic expressions of \(x\) and state the domain on which the equivalence is valid. $$ \cos (2 \arctan (x)) $$

In Exercises \(208-210\), find the two acute angles in the right triangle whose sides have the given lengths. Express your answers using degree measure rounded to two decimal places. $$ 5,12 \text { and } 13 $$
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