Problem 82 - Precalculus

Question

Does there exist an angle \(0 \leq \theta<2 \pi\) such that \(\sec \theta=\csc (-\theta) ?\)

Step-by-Step Solution

Answer

Yes, at \( \theta = \frac{3\pi}{4} \).

Step 1: Understand the Problem

We need to find if there is an angle \( \theta \) within the interval \( 0 \leq \theta < 2\pi \) such that \( \sec \theta = \csc(-\theta) \). Recall that \( \sec \theta = \frac{1}{\cos \theta} \) and \( \csc(-\theta) = \frac{1}{\sin(-\theta)} \).

Step 2: Simplify the Expression

Since \( \sin(-\theta) = -\sin(\theta) \), the expression \( \csc(-\theta) \) simplifies to \( -\csc(\theta) = -\frac{1}{\sin(\theta)} \). The original equation becomes \( \sec \theta = -\csc \theta \).

Step 3: Set Up the Equation

From Step 2, we need to solve \( \frac{1}{\cos \theta} = -\frac{1}{\sin \theta} \). Cross-multiplying gives \( \sin \theta = -\cos \theta \).

Step 4: Solve the Trigonometric Equation

Rewriting \( \sin \theta = -\cos \theta \) gives \( \tan \theta = -1 \). This equation is satisfied for angles \( \theta \) where the tangent function equals \(-1\).

Step 5: Determine Values of \( \theta \)

The tangent function equals \(-1\) at angles where the terminal side is in the second and fourth quadrants. Therefore, \( \theta = \frac{3\pi}{4} \) or \( \theta = \frac{7\pi}{4} \) within the specified interval \( 0 \leq \theta < 2\pi \).

Step 6: Verify Solutions

Check both solutions: \( \sec\left(\frac{3\pi}{4}\right) = -\sqrt{2} \) and \( \csc\left(-\frac{3\pi}{4}\right) = -\sqrt{2} \), and for \( \sec\left(\frac{7\pi}{4}\right) = \sqrt{2} \) and \( \csc\left(-\frac{7\pi}{4}\right) = -\sqrt{2} \). Only \( \theta = \frac{3\pi}{4} \) satisfies the original condition.

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is related to the cosine function. It is defined as the reciprocal of the cosine function. Formally, \( \sec \theta = \frac{1}{\cos \theta} \). This means that wherever the cosine function is zero, the secant function becomes undefined due to division by zero. The behavior of secant is important because, in trigonometry, understanding how each function behaves helps solve complex equations.

• The secant function is undefined whenever \( \cos \theta = 0 \), which happens at odd multiples of \( \frac{\pi}{2} \) or \( 90° \).
• Secant has vertical asymptotes at the same points where cosine is zero.
• The range of the secant function is all real numbers except for the interval (-1, 1), since cosine outputs values in this interval.

By knowing the properties of the secant function, you can solve equations that involve reciprocal trigonometric functions effectively.

Cosecant Function

The cosecant function is another reciprocal-based function, denoted as \( \csc \theta \) and is the reciprocal of the sine function. Thus, \( \csc \theta = \frac{1}{\sin \theta} \). Like the secant function, the cosecant becomes undefined at certain points, specifically where sine is zero.

• Cosecant is undefined at \( \theta = n\pi \), where \( n \) is any integer, because the sine function equals zero at multiples of \( \pi \).
• The range of the cosecant function follows all real numbers except the interval (-1, 1).
• It exhibits vertical asymptotes at integral multiples of \( \pi \).

In the given problem, we simplify \( \csc(-\theta) \) as \(-\csc(\theta)\) because sine is an odd function, meaning \( \sin(-\theta) = -\sin(\theta) \). This relationship plays a crucial role in simplifying trigonometric equations effectively.

Tangent Function

The tangent function, represented as \( \tan \theta \), is the ratio of the sine function to the cosine function. It is expressed as:\[\tan \theta = \frac{\sin \theta}{\cos \theta}\]This relationship is fundamental because when sine and cosine are equal in magnitude but opposite in sign, the tangent becomes -1, a key insight for solving the problem at hand.

• Tangent has vertical asymptotes where \( \cos \theta = 0 \), which corresponds to points where secant and tangent are undefined.
• The principal period of the tangent function is \( \pi \), meaning it repeats every \( \pi \) radians.
• Tangent is continuous with no breaks within its period, except where it encounters its vertical asymptotes.

The equation \( \tan \theta = -1 \) suggests that the tangent function outputs -1 at specific angles. In this case, the correct angles within the interval for \( \theta \) are \( \frac{3\pi}{4} \) and \( \frac{7\pi}{4} \), which are derived from the properties of tangent and its symmetry in different quadrants.