Problem 42 - Precalculus

Question

Use the unit circle to find all of the exact values of \(\theta\) that make the equation true in the indicated interval. $$\cos \theta=0,0 \leq \theta \leq 4 \pi$$

Step-by-Step Solution

Answer

Solutions are \( \theta = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \frac{7\pi}{2} \).

Step 1: Recognize the Trigonometric Equation

The equation to solve is \( \cos \theta = 0 \), which means we need to find all angles \( \theta \) on the unit circle where the cosine value equals zero. Cosine represents the x-coordinate on the unit circle.

Step 2: Identify Where Cosine Equals Zero

On the unit circle, \( \cos \theta = 0 \) at \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \) because these angles correspond to the points (0,1) and (0,-1) respectively, where the x-coordinate (cosine) is zero.

Step 3: Find All Solutions within the Given Interval

Starting from \( \theta = 0 \) to \( \theta = 4\pi \), we explore the solutions in placements of multiples of 2\(\pi\) (a full circle). Thus, additional solutions are \( \theta = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \frac{7\pi}{2} \). These angles are obtained by rotating 0.5 circles (or 180 degrees) from the starting point of each new circle.

Key Concepts

Trigonometric Equations

Trigonometric equations are mathematical statements that relate trigonometric functions, such as sine, cosine, and tangent, to some values within a specific interval. These equations use angles, usually represented by \(\theta\), and are typically solved over specific intervals. For example, in the provided exercise, we investigate values of \(\theta\) where the cosine function equals zero within the interval \(0 \leq \theta \leq 4\pi\). Solving trigonometric equations often involves determining where these functions take on specific values on the unit circle.

To solve such equations, we identify key points on the unit circle where the trigonometric function holds certain values. Knowing how to navigate these solutions on the unit circle is crucial. It's our main tool in finding precise angles that meet the equation's requirements. This skill is fundamental for solving various real-world problems that involve periodic and wave-like functions.

Cosine Function

The cosine function, denoted as \(\cos \theta\), corresponds to the x-coordinate of a point on the unit circle for any angle \(\theta\). The unit circle is a circle of radius one, centered at the origin of a coordinate plane. This function is periodic with a period of \(2\pi\), meaning its values repeat every \(2\pi\) units of angle.

In the context of the unit circle, the cosine function takes value zero at specific angles where the circle's corresponding point lies vertically aligned on the y-axis, either at the top (0, 1) or bottom (0, -1). For this reason, \(\cos \theta = 0\) at \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\). Understanding this property helps in solving trigonometric equations and is useful across various mathematical and engineering fields.

Angle Intervals

Understanding angle intervals is crucial for solving trigonometric equations. An angle interval sets the bounds within which we search for solutions. In our exercise, the interval \(0 \leq \theta \leq 4\pi\) means we are looking for all angles between 0 and \(4\pi\) radians where \(\cos \theta = 0\).

Working within a given interval ensures we find all necessary solutions. In this case, the process involves not just noting where \(\cos \theta = 0\) within one complete cycle \(0 \leq \theta \leq 2\pi\), but continuing the search across additional cycles until we reach \(4\pi\). This problem illustrates the periodic nature of trigonometric functions, emphasizing why angle intervals are integral to solving these problems correctly.