Problem 33 - Precalculus

Question

Find the indicated trigonometric function values. If \(\csc \theta=\frac{2}{\sqrt{3}}\) and the terminal side of \(\theta\) lies in quadrant II, find \(\cot \theta\)

Step-by-Step Solution

Answer

\( \cot \theta = -\frac{\sqrt{3}}{3} \)

Step 1: Understand the given information

We are given that \( \csc \theta = \frac{2}{\sqrt{3}} \). The cosecant function is the reciprocal of the sine function. Thus, \( \sin \theta = \frac{\sqrt{3}}{2} \). Additionally, we know that \( \theta \) is in the second quadrant.

Step 2: Determine the angle's sign behavior in Quadrant II

In the second quadrant, \( \sin \theta \) remains positive and \( \cos \theta \) is negative. Therefore, \( \cos \theta = -\sqrt{1 - \sin^2 \theta} \) since cosine values are negative in Quadrant II.

Step 3: Calculate \( \cos \theta \)

Use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \sin \theta = \frac{\sqrt{3}}{2} \) to find \( \cos \theta = -\sqrt{1 - \left(\frac{\sqrt{3}}{2}\right)^2} = -\sqrt{1 - \frac{3}{4}} = -\sqrt{\frac{1}{4}} = -\frac{1}{2} \).

Step 4: Finding \( \cot \theta \)

Recall that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Substitute the values we found: \( \cos \theta = -\frac{1}{2} \) and \( \sin \theta = \frac{\sqrt{3}}{2} \). Thus, \( \cot \theta = \frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} \).

Key Concepts

Cosecant

Cosecant is one of the six fundamental trigonometric functions, and it is particularly important when dealing with right-angle triangles or the unit circle. It's often denoted as \( \csc \theta \), which stands for cosecant of an angle \( \theta \). It is the reciprocal of the sine function.

• If \( \, \sin \theta = a \), then \( \csc \theta = \frac{1}{a} \).
• To find \( \, \sin \theta \) when given \( \csc \theta \), take the reciprocal of the value.

In practical scenarios, knowing \( \csc \theta \) provides direct insight into the hypotenuse and opposite side length ratios. In this example, we had \( \csc \theta = \frac{2}{\sqrt{3}} \), meaning the sine value, which is the reciprocal, becomes \( \sin \theta = \frac{\sqrt{3}}{2} \). Understanding cosecant is crucial because it helps determine other trigonometric values by connecting these relationships.

Cotangent

Cotangent is another fundamental trigonometric function represented as \( \cot \theta \). It is defined as the reciprocal of the tangent function:

• \( \cot \theta = \frac{1}{\tan \theta} \).
• Alternatively, it can be expressed as \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

Cotangent signifies the ratio of the adjacent side to the opposite side in a right-angled triangle, helping solve problems where angle measures and side lengths involve such ratios. In this question, after calculating \( \cos \theta = -\frac{1}{2} \) and knowing \( \sin \theta = \frac{\sqrt{3}}{2} \), we determined that \( \cot \theta = \frac{\cos \theta}{\sin \theta} = -\frac{\sqrt{3}}{3} \). Being familiar with cotangent is useful for quadrant analysis, especially when identifying angle behaviors.

Quadrant Analysis

Quadrant analysis is essential for understanding how trigonometric functions behave in different regions of the coordinate plane. The plane is divided into four quadrants, each affecting trigonometric functions uniquely:

• Quadrant I: All trigonometric functions are positive.
• Quadrant II: Sine is positive; cosine and tangent are negative.
• Quadrant III: Tangent is positive; sine and cosine are negative.
• Quadrant IV: Cosine is positive; sine and tangent are negative.

In this exercise, knowing that the terminal side of \( \theta \) is in Quadrant II helps determine that \( \sin \theta \) is positive, while \( \cos \theta \) and hence \( \cot \theta \) are negative. This information guided us in effectively analyzing the position of angles and calculating accurate trigonometric values.