Problem 38 - Precalculus: Functions and Graphs

Question

Find the solutions of the equation that are in the interval \([0,2 \pi).\) \(\cos 2 \theta-\tan \theta=1\)

Step-by-Step Solution

Answer

No solution lies within the interval [0, 2π) for the given equation.

Step 1: Double Angle Identity

Using the double angle identity for cosine, we can rewrite \( \cos 2\theta \) as \( 2\cos^2 \theta - 1 \). This transforms the equation \( \cos 2\theta - \tan \theta = 1 \) into \( 2\cos^2 \theta - 1 - \tan \theta = 1 \).

Step 2: Simplify Equation

Substituting \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), we rewrite the equation as \( 2\cos^2 \theta - 1 - \frac{\sin \theta}{\cos \theta} = 1 \). This simplifies the expression by expressing all terms in sine and cosine.

Step 3: Rearrange Terms

To simplify further, we rearrange the terms: \( 2\cos^2 \theta - 2 - \frac{\sin \theta}{\cos \theta} = 0 \). To make it easier to solve, isolate the fraction: \( 2\cos^2 \theta - 2 = \frac{\sin \theta}{\cos \theta} \).

Step 4: Solve for Sine and Cosine

Use substitution or trigonometric identities to solve the new equation for \( \sin \theta \) and \( \cos \theta \). Sometimes guessing based on common angles (like \( \frac{\pi}{6}, \frac{\pi}{4} \)) that satisfy \( \tan \theta = \frac{\sin \theta}{\cos \theta} \approx 0 \) can help.

Step 5: Check Validity

Using angles like \( \theta = 0, \theta = n\pi \) (where \( n \) is an integer) check which satisfy the entire equation \( \cos 2\theta - \tan \theta = 1 \). Evaluate \( \cos 2\theta \) and \( \tan \theta \) accordingly.

Step 6: Consider Interval

Ensure that any solution found in previous steps lies within the interval \([0, 2\pi)\). Discard any solution that is not within this range.

Key Concepts

Double Angle Identity

When solving trigonometric equations like \(\cos 2\theta - \tan \theta = 1\), utilizing the double angle identity can be extremely helpful. The double angle identities are specific trigonometric formulas that express trigonometric functions of double angles \(2\theta\) in terms of single angles \(\theta\). Here we use the identity for cosine:

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

Substituting this identity into the equation helps simplify it by reducing the complexity of the terms involved. With the double angle identity, our equation becomes \(2\cos^2 \theta - 1 - \tan \theta = 1\). This transformed equation is easier to manipulate and set up for further simplification. The use of identities is a crucial skill in solving all types of trigonometric equations, allowing for transformations that bring us closer to a solvable equation.

Sine and Cosine

In trigonometric equations like \(2\cos^2 \theta - 1 - \tan \theta = 1\), breaking terms into sine and cosine can help find a solution. Tan can be expressed in terms of sine and cosine as:

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

Substituting \(\tan \theta\) with \(\frac{\sin \theta}{\cos \theta}\) turns all terms into single-angle equations that involve sine or cosine. This might seem more complicated at first, but it provides a clear path for solving or manipulating the equation further.
Sometimes finding exact solutions for \(\theta\) in simplified trigonometric equations involves recognizing common angles that have known sine and cosine values, like \(\theta = \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}\). Trigonometric reference angles help in quickly identifying possible solutions without having to calculate each one manually.

Interval Solution

Finding solutions to a trigonometric equation is often only the first step. The next crucial step is ensuring that these solutions fit within the given interval, which in this case is \([0, 2\pi)\). This interval represents one full cycle of the angular functions.
When solutions are obtained, we must check each one against this interval. Sometimes, solutions may stem from multiples of \(\pi\) or expressions outside the interval, such as negative angles. It's important to:

• Convert any angle beyond \(2\pi\) to its equivalent within the interval by subtracting \(2\pi\).
• Ignore any negative or non-qualifying solutions that fall outside \([0, 2\pi)\).

This ensures that we have a valid, complete set of solutions that adhere to the constraints of the problem. Observing these boundaries allows us to refine our solutions and ensures they make sense within the context of the problem, where angles are typically restricted to one complete cycle of the unit circle.