Problem 13 - Algebra 2

Question

\(\cot \theta=-\frac{\sqrt{2}}{2},\) Quadrant \(\mathrm{IV}\)

Step-by-Step Solution

Answer

\( \theta = 315^\circ \) or \( \theta = \frac{7\pi}{4} \).

Step 1: Understanding the Relationship

The cotangent of an angle is defined as the ratio of the adjacent side to the opposite side in a right triangle. Therefore, \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). In Quadrant IV, \( \cos \theta \) is positive and \( \sin \theta \) is negative.

Step 2: Expressing Cotangent in Terms of Sine and Cosine

Since \( \cot \theta = -\frac{\sqrt{2}}{2} \), it follows that \( \frac{\cos \theta}{\sin \theta} = -\frac{\sqrt{2}}{2} \). We need to find values of \( \cos \theta \) and \( \sin \theta \) that satisfy this equation while abiding by the sign properties in Quadrant IV.

Step 3: Determine \(\cos \theta\) and \(\sin \theta\)

Since \( \cot \theta = -\frac{\sqrt{2}}{2} \), and considering the known triangles and angles, we find that \( \cos \theta = \frac{1}{2} \sqrt{3} \) and \( \sin \theta = -\frac{1}{2} \) satisfy both the trigonometric identity \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) and the sign rules for Quadrant IV angles.

Step 4: Identify the Angle \(\theta\)

The coordinates found correspond to a specific angle. Since \( \cos \theta \) is positive and \( \sin \theta \) is negative in Quadrant IV, the angle \( \theta = 315^\circ \) (or in radians, \( \frac{7\pi}{4} \)).

Key Concepts

Cotangent

The cotangent function, often denoted as \( \cot \theta \), is a trigonometric ratio that compares the adjacent side to the opposite side in a right triangle. You can express it in terms of sine and cosine as:

• \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

This is essential for determining angles and sides in various mathematical problems.
When dealing with negative cotangent values, like the one in our problem \( \cot \theta = -\frac{\sqrt{2}}{2} \), it indicates a relationship between the cosine and sine of the angle that involves opposite signs. Understanding this property helps in identifying which quadrant the angle resides in. Quadrants affect the sign of these trigonometric functions, therefore guiding you to the appropriate angle measurements.

Quadrant IV

In trigonometry, the unit circle is divided into four quadrants to help determine the signs of trigonometric functions. Quadrant IV is particularly unique:

• In Quadrant IV, \( \cos \theta \) is positive, while \( \sin \theta \) is negative.
• Important to recognize that this quadrant lies between the angles \( 270^\circ \) and \( 360^\circ \).

Finding\( \theta \)
For the case mentioned, we needed to find an angle where \( \cot \theta \) is negative. Since \( \cos \theta = \frac{\sqrt{3}}{2} \) and \( \sin \theta = -\frac{1}{2} \) matches these sign rules, it helps us deduce that the angle \( \theta \) is \( 315^\circ \) (or \( \frac{7\pi}{4} \) radians). This understanding of Quadrant IV aids in locating precise angles quickly.

Trigonometric Identities

Trigonometric identities are powerful tools that relate different trigonometric functions to one another. Being familiar with these identities can simplify complex problems. One key identity involving cotangent is:

• \( \cot \theta = \frac{1}{\tan \theta} \)

Useful transformations, like \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), allow easier calculation by breaking the problem into smaller manageable parts.
In our exercise, using these identities allowed us to change the problem into something more familiar, by starting with the given \( \cot \theta = -\frac{\sqrt{2}}{2} \). We deduced \( \cos \theta = \frac{1}{2} \sqrt{3} \) and \( \sin \theta = -\frac{1}{2} \) aligning with those trigonometric identities and matching the signs appropriate for Quadrant IV. Mastery over these identities is vital for solving trigonometric problems efficiently.