Problem 36 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximale values in radians to four decimal places and approximate values in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$4 \cos 2 \theta=8 \sin \theta \cos \theta$$

Step-by-Step Solution

Answer

The solutions are \( \theta = 22.5^\circ, 112.5^\circ, 202.5^\circ, 292.5^\circ \).

Step 1: Simplify the equation

First, observe the equation: \(4 \cos 2\theta = 8 \sin \theta \cos \theta\). We can simplify the right-hand side using the double-angle identity for sine: \(\sin 2\theta = 2\sin \theta \cos \theta\). This allows us to rewrite the equation as \(4 \cos 2\theta = 4 \sin 2\theta\).

Step 2: Cancel common factors

Both sides of the equation, \(4 \cos 2\theta = 4 \sin 2\theta\), can be divided by 4, assuming \(\cos 2\theta\) and \(\sin 2\theta\) are also non-zero. This simplifies the equation to \(\cos 2\theta = \sin 2\theta\).

Step 3: Express the equation in terms of tangent

Divide both sides by \(\cos 2\theta\) to yield: \(1 = \tan 2\theta\). This implies \(2\theta = k\pi + \frac{\pi}{4}\), where \(k\) is an integer, because the tangent function repeats every \(\pi\) radians.

Step 4: Solve for \(\theta\)

Solve for \(\theta\) by dividing each term by 2: \[\theta = \frac{k\pi}{2} + \frac{\pi}{8}\]. This gives the general solution for \(\theta\) in radians.

Step 5: Convert radians to degrees

Convert the general solution to degrees using the conversion \(180^\circ = \pi\): \[\theta = \left(\frac{k\pi}{2} + \frac{\pi}{8}\right) \times \frac{180^\circ}{\pi} = 90k^\circ + 22.5^\circ \].

Step 6: Determine nonnegative solutions

To find the nonnegative angles less than 360° in degrees, solve \(90k^\circ + 22.5^\circ = 22.5^\circ, 112.5^\circ, 202.5^\circ, 292.5^\circ\) by choosing appropriate integer values of \(k = 0, 1, 2, 3\).

Key Concepts

Double-Angle Identities

Understanding the double-angle identities is crucial when solving trigonometric equations such as the one in this exercise. These identities simplify expressions involving trigonometric functions of double angles, making it easier to solve equations. For instance, the double-angle identity for sine is given by \( \sin 2\theta = 2 \sin \theta \cos \theta \). This identity can help convert complex equations into simpler forms by relating the functions of double angles back to single angles.

Using double-angle identities, we can transform complex trigonometric expressions into simpler ones. In our example exercise, the equation \( 4 \cos 2\theta = 8 \sin \theta \cos \theta \) was simplified to \( 4 \cos 2\theta = 4 \sin 2\theta \) using the identity \( \sin 2\theta = 2 \sin \theta \cos \theta \). This simplification is vital in finding the solutions to the problem more efficiently.

Solving Trigonometric Equations

Solving trigonometric equations involves several steps, and understanding each step is essential for mastering the process. First, it is crucial to simplify the equation using identities or algebraic manipulation, if possible. In the exercise given, the equation was simplified from \( 4 \cos 2\theta = 8 \sin \theta \cos \theta \) to \( \cos 2\theta = \sin 2\theta \).

Once simplified, the next goal is to isolate the trigonometric function. In this case, dividing both sides by \( \cos 2\theta \) reduced the equation to \( 1 = \tan 2\theta \). By doing so, the problem became one of solving for an angle whose tangent is 1, which is more straightforward.

The solutions are often expressed using general formulas because trigonometric functions are periodic. For the tangent function, solutions occur at intervals of \( \pi \) radians because \( \tan(\theta) = \tan(\theta + n\pi) \) for any integer \( n \). This knowledge is applied to find the most general solutions possible.

Conversion Between Radians and Degrees

Working with both radians and degrees is a common requirement when dealing with trigonometric equations. Whether a problem demands a specific unit or requires conversion, understanding this conversion is important. The conversion factor between radians and degrees is that \( \pi \) radians are equivalent to \( 180^\circ \).

When converting from radians to degrees, multiply the radians by \( \frac{180}{\pi} \). Conversely, convert degrees to radians by multiplying the degrees by \( \frac{\pi}{180} \). In the exercise, the general solution \( \theta = \frac{k\pi}{2} + \frac{\pi}{8} \) in radians was converted to degrees as \( \theta = 90k^\circ + 22.5^\circ \). This conversion allows you to work with the unit that best fits the problem's requirements or the unit you are most comfortable with.

General Solution for Trigonometric Equations

Finding the general solution for trigonometric equations involves considering the periodic nature of trigonometric functions. Because these functions repeat at regular intervals, solutions to equations can often be expressed in terms of a variable \( k \), which represents an integer. This captures all possible solutions succinctly.

In this problem, the equation initially reduced to \( 1 = \tan 2\theta \), which translates to \( 2\theta = k\pi + \frac{\pi}{4} \). Dividing through by 2 yields \( \theta = \frac{k\pi}{2} + \frac{\pi}{8} \), which is the general solution in radians.

The idea here is to understand that for every \( k \), the equation gives another valid angle, illustrating the infinite solutions that arise from the periodicity of trigonometric functions. Hence, when you specify the general solution, you account for all possible scenarios derived from multiples of the function's period.