Problem 29 - Precalculus Mathematics for Calculus

Question

An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0,2 \pi)\) $$\sec \theta-\tan \theta=\cos \theta$$

Step-by-Step Solution

Answer

Solutions in [0, 2π) are θ = 0, π/2, π.

Step 1: Express Secant and Tangent in Terms of Sine and Cosine

The given equation is \(\sec \theta - \tan \theta = \cos \theta\). We start by expressing \(\sec \theta\) and \(\tan \theta\) in terms of sine and cosine:\[\sec \theta = \frac{1}{\cos \theta}\]\[\tan \theta = \frac{\sin \theta}{\cos \theta}\]So the equation becomes:\[\frac{1}{\cos \theta} - \frac{\sin \theta}{\cos \theta} = \cos \theta\]

Step 2: Simplify the Left Side of the Equation

Combine the terms on the left by using the common denominator \(\cos \theta\):\[\frac{1 - \sin \theta}{\cos \theta} = \cos \theta\]Now multiply both sides by \(\cos \theta\) to eliminate the fraction:\[1 - \sin \theta = \cos^2 \theta\]

Step 3: Use the Pythagorean Identity to Simplify

Recall the Pythagorean identity \(\cos^2 \theta = 1 - \sin^2 \theta\). Substitute this into the equation:\[1 - \sin \theta = 1 - \sin^2 \theta\]Cancel the \(1\) on both sides:\[-\sin \theta = -\sin^2 \theta\]Divide through by \(-1\) to simplify:\[\sin \theta = \sin^2 \theta\]

Step 4: Solve the Resulting Quadratic Equation

The equation \(\sin \theta = \sin^2 \theta\) can be rearranged to:\[\sin^2 \theta - \sin \theta = 0\]Factor out \(\sin \theta\):\[\sin \theta (\sin \theta - 1) = 0\]This gives two solutions:\(\sin \theta = 0\) or \(\sin \theta = 1\).

Step 5: Determine All Solutions

For \(\sin \theta = 0\), \(\theta = n\pi\) where \(n\) is an integer.For \(\sin \theta = 1\), \(\theta = \frac{\pi}{2} + 2k\pi\) where \(k\) is any integer.

Step 6: Find Solutions in the Interval [0, 2π)

We only consider \(\theta\) values in the interval \([0, 2\pi)\):- For \(\sin \theta = 0\): \(\theta = 0\) and \(\theta = \pi\).- For \(\sin \theta = 1\): \(\theta = \frac{\pi}{2}\).Therefore, the solutions in the interval \([0, 2\pi)\) are \(\theta = 0, \frac{\pi}{2}, \pi\).

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is a trigonometric function that is the reciprocal of the cosine function. This means that the secant of an angle \( \theta \) is defined as the ratio of 1 to the cosine of \( \theta \). In mathematical terms, this is written as:

• \( \sec \theta = \frac{1}{\cos \theta} \)

Understanding the secant function helps in solving trigonometric equations where cosines are involved.
It often leads equations into a form that can be more easily manipulated or simplified. When working with secant in the context of trigonometric identities or equations, always remember to consider the behavior of the cosine function. Since secant is undefined wherever cosine is zero, you must take care to identify these points, as they play a crucial role in solving equations.

Tangent Function

The tangent function, represented by \( \tan \theta \), is another basic trigonometric function. It relates the sine and cosine functions through the following relationship:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

The tangent function is particularly useful because it provides a direct comparison between the sine and cosine functions.
An important aspect to note is that the tangent function has points of discontinuity, which occur wherever the cosine function is zero (since dividing by zero is undefined).
In solving trigonometric equations, expressing terms in a common denominator, as illustrated by converting secant and tangent to sine and cosine terms, is often a key strategic step.
This can simplify the process of combining terms and solving for the variable \( \theta \).

Pythagorean Identity

The Pythagorean identities are fundamental relationships in trigonometry. The one most used in this context is:

• \( \cos^2 \theta + \sin^2 \theta = 1 \)

This identity helps to simplify expressions and solve equations where both sine and cosine functions appear. One can rearrange this identity to express \( \cos^2 \theta \) in terms of \( \sin^2 \theta \) as follows:

• \( \cos^2 \theta = 1 - \sin^2 \theta \)

Using this identity allows us to transform terms in the equations, making it easier to solve for known values like in the given example.
By substituting \( \cos^2 \theta \) with \( 1 - \sin^2 \theta \), you can connect the dots between different trigonometric elements and arrive at solutions methodically.
Understanding and applying these identities can simplify complex problems into solvable equations, especially when factoring and solving quadratic-like trigonometric equations.