Problem 44 - Precalculus

Question

Use the unit circle to find all of the exact values of \(\theta\) that make the equation true in the indicated interval. $$\cot \theta=1,0 \leq \theta \leq 2 \pi$$

Step-by-Step Solution

Answer

\(\theta = \frac{\pi}{4}, \frac{5\pi}{4}\).

Step 1: Understanding Cotangent

The cotangent function is the reciprocal of the tangent function. Hence, if \( \cot \theta = 1 \), then \( \frac{\cos \theta}{\sin \theta} = 1 \), which means \( \cos \theta = \sin \theta \). To find \( \theta \), we look for angles where the sine and cosine values are equal on the unit circle.

Step 2: Identify Key Points on the Unit Circle

On the unit circle, \( \sin \theta = \cos \theta \) at angles where the coordinates are \( (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) \). In the interval \( 0 \leq \theta \leq 2\pi \), these angles are \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{5\pi}{4} \).

Step 3: Verify the Interval

Both \( \frac{\pi}{4} \) and \( \frac{5\pi}{4} \) fall within the interval \( 0 \leq \theta \leq 2\pi \). Thus, these are the correct solutions for \( \theta \).

Step 4: Conclusion

The angles \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{5\pi}{4} \) make the equation \( \cot \theta = 1 \) true within the specified interval, as at these points, \( \sin \theta = \cos \theta \).

Key Concepts

Understanding the Cotangent Function

The cotangent function, often abbreviated as \( \cot \theta \), is a trigonometric function that is the reciprocal of the tangent function. In mathematical terms, this means that \( \cot \theta = \frac{1}{\tan \theta} \). Since the tangent of an angle \( \theta \) is defined as the ratio of the sine and cosine of the angle, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), we can express the cotangent as \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

Understanding cotangent helps us solve equations involving this function by converting them into more familiar terms involving sine and cosine. In our exercise, if \( \cot \theta = 1 \), it follows that \( \frac{\cos \theta}{\sin \theta} = 1 \), implying \( \cos \theta = \sin \theta \). This gives us the points on the unit circle where the sine and cosine values must be equal.

Exploring Common Trigonometric Functions

Trigonometric functions are fundamental to understanding the relationships between angles and the sides of a triangle, especially in a unit circle setting. The main trigonometric functions include sine \( \sin \), cosine \( \cos \), tangent \( \tan \), cotangent \( \cot \), secant \( \sec \), and cosecant \( \csc \).

The unit circle is a powerful tool in trigonometry, as it provides a visual representation of these functions. It is a circle with a radius of one, centered at the origin of the coordinate plane. Sine and cosine represent the y- and x-coordinates of a point on this circle, respectively. Thus, as an angle \( \theta \) increases or decreases, its sine and cosine values change, which directly affects the values of tangent and cotangent. By visualizing the unit circle, it becomes straightforward to understand the periodic nature and relationships of these functions.

For \( \sin \theta = \cos \theta \), we specifically look at the unit circle to identify angles where this condition is met, which correspond to the points where the line \( y = x \) intersects the circle.

Finding Exact Values on the Unit Circle

To find exact values for trigonometric functions, especially within specific intervals, the unit circle serves as an essential tool. Locations on this circle correspond to specific angles, and their coordinates directly give sine and cosine values.

In our case, finding \( \theta \) such that \( \cot \theta = 1 \) requires us to find angles where \( \sin \theta = \cos \theta \). On the unit circle, this occurs where the points are symmetrically placed relative to the line \( y = x \), particularly at angles \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{5\pi}{4} \). Both these specific angles, measured in radians, provide exact values due to their familiar coordinates \( (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) \), representing a 45-degree (or \( \frac{\pi}{4} \) radians) angle in various quadrants of the circle.

By knowing these key intersections and their exact coordinates, you can confidently solve equations involving any trigonometric functions, using the unit circle as a reliable reference.