Problem 34 - Algebra and Trigonometry

Question

Involve equations with multiple angles. Solve each equation on the interval \([0,2 \pi)\) $$ \cos \frac{2 \theta}{3}=-1 $$

Step-by-Step Solution

Answer

The solution to the equation \( \cos \frac{2 \theta}{3}=-1 \) on the interval \([0,2 \pi)\) is \( \theta = \frac{3\pi}{2} \).

Step 1: Identify the cosine function

The first step is to identify when cosine function equals -1. \( \cos x = -1 \) for \( x = \pi \). Therefore, \( \frac{2 \theta}{3} = \pi \).

Step 2: Solve for theta

The next step is to solve the equation for \( \theta \). Multiplying both sides by \(\frac{3}{2}\), we get \( \theta = \frac{3\pi}{2} \).

Step 3: Check that the solution is in the required interval

Lastly, we must verify that the obtained solution lies within the given interval \([0,2 \pi)\). \( \frac{3\pi}{2} \) indeed lies within this interval.

Key Concepts

Cosine Function

The cosine function is one of the primary trigonometric functions and captures the relationship between the angles and sides of a right-angled triangle. Specifically, when given an angle in standard position, it represents the horizontal coordinate of a point where the terminal side of the angle intersects the unit circle. Visually, if you wrapped the unit circle on a coordinate plane around a right triangle, the cosine of the angle would align with the adjacent side over the hypotenuse, hence the basic trigonometric identity \( \cos(\theta) = \frac{adjacent}{hypotenuse} \).

For the given exercise \( \cos \frac{2 \theta}{3}=-1 \), it is crucial to recognize that the cosine function takes on the value of -1 at an angle of \( \pi \) radians (or 180 degrees), which occurs at the negative x-axis on the unit circle. This recognition is essential in trigonometric equations as it guides us to the specific angles where the cosine of that angle yields the required value, which is the backbone of solving such equations.

Multiple Angle Formulas

Multiple angle formulas are a part of trigonometry that express trigonometric functions of multiples of angles, such as twice or three times an angle, in terms of the trigonometric functions of the original angle. These formulas are indispensable when dealing with trigonometric equations that involve angles multiplied by a constant.

For example, the multiple angle formula for cosine is \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \), which can be further transformed using the Pythagorean identity into \( \cos(2\theta) = 2\cos^2(\theta) - 1 \) or \( \cos(2\theta) = 1 - 2\sin^2(\theta) \). In our exercise, we have a multiple of \( \frac{2}{3} \) which means we need to adjust this formula accordingly. Although the given problem doesn't require the use of the formula counterpart due to the straight forward nature of the cosine value (-1), understanding the presence of such a formula assists greatly in solving more complicated multiple angle scenarios.

Angle Measurement

Angle measurement in trigonometry can typically be done in radians or degrees. Understanding both systems is essential for solving trigonometric equations. The radian measure is especially important because it provides a direct connection between arc length and radius, leading to a number of important calculus concepts. One full revolution around a circle corresponds to \(2\pi\) radians or 360 degrees.

In the provided exercise, the interval is given in radians, from 0 to \(2\pi\). It is important to ensure that solutions fall within this interval because trigonometric functions are periodic in nature; they repeat values over intervals. For instance, \(\cos(\theta)\) is the same as \(\cos(\theta + 2\pi)\), but the latter is not within our defined interval. Hence, when solving \( \cos \frac{2 \theta}{3}=-1 \) and obtaining \( \theta = \frac{3\pi}{2} \), we must check that \( \frac{3\pi}{2} \) is within our interval, which in this case, it is, since it falls between 0 and \(2\pi\).