Problem 27 - Algebra and Trigonometry

Question

\(17-34\) . An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0,2 \pi) .\) $$ \sin 2 \theta=3 \cos 2 \theta $$

Step-by-Step Solution

Answer

Solutions in \([0, 2\pi)\) are \(\theta \approx 0.624, 2.195, 3.715, 5.285\).

Step 1: Begin with the given equation

Start by rearranging the given equation \( \sin 2\theta = 3 \cos 2\theta \) into a single trigonometric function. To do this, divide both sides by \( \cos 2\theta \), assuming \( \cos 2\theta eq 0 \). This gives \( \tan 2\theta = 3 \).

Step 2: Solve for \(2\theta\)

To find \(2\theta\), we take the arctan of both sides: \( 2\theta = \tan^{-1}(3) \). This gives the principal solution. Remember that the tangent function is periodic with period \(\pi\), so \( 2\theta = \tan^{-1}(3) + n\pi \), where \(n\) is any integer.

Step 3: Solve for \(\theta\)

Divide the equation from Step 2 by 2 to solve for \( \theta \). Thus, \( \theta = \frac{1}{2} \tan^{-1}(3) + \frac{n\pi}{2} \).

Step 4: Find solutions in the interval \([0, 2\pi)\)

To find the solutions within \([0, 2\pi)\), substitute different integer values of \(n\) into the expression for \(\theta\) and calculate if \(\theta\) remains within the desired interval.Calculate:- For \(n = 0\), \(\theta = \frac{1}{2} \tan^{-1}(3)\)- For \(n = 1\), \(\theta = \frac{1}{2} \tan^{-1}(3) + \frac{\pi}{2}\)- For \(n = 2\), \(\theta = \frac{1}{2} \tan^{-1}(3) + \pi\)- Continue substituting subsequent integers for \(n\) until \(\theta\) exceeds \(2\pi\).Only retain those \(\theta\) values that fall within \([0, 2\pi)\).

Step 5: Verify Integer Solutions

Compute the actual values for \( \tan^{-1}(3) \) (approximately 1.249) to find numerical solutions for \( \theta \):- For \(n = 0\), \(\theta \approx 0.624\)- For \(n = 1\), \(\theta \approx 2.195\)- For \(n = 2\), \(\theta \approx 3.715\)- For \(n = 3\), \(\theta \approx 5.285\)Confirm these are within the interval \([0, 2\pi)\).

Key Concepts

Trigonometric Identities

Trigonometric identities are powerful tools in solving equations that involve trigonometric functions. These identities allow us to simplify and manipulate trigonometric expressions. In this exercise, we start by using the identity for tangent, which relates sine and cosine:

• \( \tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta} \)

This identity is instrumental because it helps us convert the original equation, \(\sin 2\theta = 3 \cos 2\theta\), into a form where we can solve for \(\theta\) more easily. By dividing both sides by \(\cos 2\theta\), provided \(\cos 2\theta eq 0\), we convert the equation into \(\tan 2\theta = 3\). Trigonometric identities thus help us bridge the gap between complex expressions and simpler equations.

Solutions to Equations

Solving trigonometric equations often involves transforming them into a standard form where solutions are more readily found. After transforming the given trigonometric equation into \( \tan 2\theta = 3 \), we find the solutions for \(2\theta\) by taking the arctangent on both sides. The solution for the tangent function is given by:

• \( 2\theta = \tan^{-1}(3) + n\pi \)

Here, \(n\) is an integer, representing the periodic nature of the tangent function, which repeats every \(\pi\). Thus, the solutions form a series based on \(n\), and by dividing through by 2, we find \( \theta = \frac{1}{2}\tan^{-1}(3) + \frac{n\pi}{2} \). This set describes all possible solutions to the equation.

Interval Notation

Interval notation is a mathematical shorthand used to describe a range of values, typically on a number line. In this context, we are tasked with finding solutions for \(\theta\) within the interval \([0, 2\pi)\). This range specifies that \(\theta\) should be greater than or equal to 0 and less than but not equal to \(2\pi\).

• For \(n = 0\), the calculation yields \(\theta \approx 0.624\)
• For \(n = 1\), \(\theta \approx 2.195\)
• For \(n = 2\), \(\theta \approx 3.715\)
• Finally, for \(n = 3\), \(\theta \approx 5.285\)

These solutions are checked against the interval to ensure they are valid, helping us understand the behavior of our function in a specific domain.

Tangent Function

The tangent function—\(\tan\)—is one of the principal trigonometric functions and is defined as \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). This function is unique because it repeats every \(\pi\), which is half the period of the sine and cosine functions. As such, when solving equations involving tangent, solutions occur more frequently within any given interval.

• The general solution for \(\tan\) when equated is given by \( \theta = \tan^{-1}(k) + n\pi \), where \(n\) is any integer.

In this exercise, we utilize this periodicity and key characteristic to solve \(\tan 2\theta = 3\). Understanding the tangent function's behavior allows us to predict and calculate further solutions by exploiting its periodic nature.