Problem 13 - Precalculus: Functions and Graphs

Question

Find the exact value. (a) \(\cot 120^{\circ}\) (b) \(\cot \left(-150^{\circ}\right)\)

Step-by-Step Solution

Answer

(a) \(\cot 120^{\circ} = -\frac{\sqrt{3}}{3}\); (b) \(\cot (-150^{\circ}) = \sqrt{3}\).

Step 1: Understanding Cotangent Concept

Recall that the cotangent of an angle is the reciprocal of the tangent. Therefore, for any angle \( \theta \), the cotangent is given by: \[ \cot \theta = \frac{1}{\tan \theta} \]. Additionally, the cotangent in terms of sine and cosine is \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

Step 2: Converting to Reference Angles (120°)

The angle \( 120^{\circ} \) is in the second quadrant. The reference angle is \( 180^{\circ} - 120^{\circ} = 60^{\circ} \). In the second quadrant, cotangent is negative because cosine is negative and sine is positive.

Step 3: Calculating \( \cot 120^{\circ} \)

Using the reference angle \( 60^{\circ} \), where \( \cot 60^{\circ} = \frac{1}{\sqrt{3}} \):\[ \cot 120^{\circ} = -\cot 60^{\circ} = -\frac{1}{\sqrt{3}} \]Simplifying gives \( \cot 120^{\circ} = -\frac{\sqrt{3}}{3} \).

Step 4: Converting to Reference Angles (-150°)

The angle \( -150^{\circ} \) can be written as \( 360^{\circ} - 150^{\circ} = 210^{\circ} \). The reference angle is \( 210^{\circ} - 180^{\circ} = 30^{\circ} \). In the third quadrant, cotangent is positive because both sine and cosine are negative.

Step 5: Calculating \( \cot (-150^{\circ}) \)

Using the reference angle \( 30^{\circ} \), where \( \cot 30^{\circ} = \sqrt{3} \):\[ \cot (-150^{\circ}) = \cot 210^{\circ} = \cot 30^{\circ} = \sqrt{3} \].

Step 6: Summarizing Results

The exact values obtained are:\( \cot 120^{\circ} = -\frac{\sqrt{3}}{3} \), and \( \cot (-150^{\circ}) = \sqrt{3} \).

Key Concepts

Reference Angles

Reference angles play an essential role in trigonometry, making it easier to find trigonometric values of angles beyond the first quadrant. What is a Reference Angle?
A reference angle is the smallest angle that an original angle makes with the x-axis, essentially serving as a ‘mirror’ to simplify calculations. Calculating Reference Angles:

• For angles in the second quadrant, subtract the angle from 180°.
• For angles in the third quadrant, subtract 180° from the angle.
• For angles in the fourth quadrant, subtract the angle from 360°.

These calculations are crucial when working with positive and negative angles, as demonstrated in finding the cotangent of 120° and -150°. By using reference angles, you capitalize on known values like those from well-known angles (30°, 45°, 60°), which can be really helpful.

Trigonometric Identities

Trigonometric identities are foundational tools in trigonometry, providing relationships between sine, cosine, and cotangent functions. Cotangent and Tangent:
One key identity to remember is that cotangent is the reciprocal of tangent:\[ \cot \theta = \frac{1}{\tan \theta} \]This relationship allows you to convert problems in terms of tangent to cotangent and vice versa.
Cotangent in Terms of Sine and Cosine:
Another crucial identity is expressing cotangent using sine and cosine:\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]These identities helped us find the cotangent of angles like 120° by understanding the cosine and sine values based on reference angles. Using these identities helps simplify finding exact trigonometric values.

Unit Circle

The unit circle is a vital concept in trigonometry, often serving as the key reference for angles and their corresponding trigonometric values. Understanding the Unit Circle:
The unit circle is a circle centered at the origin of a coordinate plane with a radius of 1. Every point on this circle has a coordinate \((\cos \theta, \sin \theta)\), where theta is the angle formed with the positive x-axis.
Unit Circle and Quadrants:
The unit circle helps visualize angles and their signs in different quadrants:

• First Quadrant: All trigonometric values are positive.
• Second Quadrant: Sine is positive, cosine and cotangent are negative.
• Third Quadrant: Both sine and cosine are negative, making tangent and cotangent positive.
• Fourth Quadrant: Cosine is positive, sine and tangent are negative.

When calculating cotangent for angles like -150°, you can visualize how each angle lies within the corresponding quadrant and use the properties of the unit circle to deduce the cotangent value.