Problem 52 - Precalculus

Question

Determine whether each statement is possible or not. $$\cot \theta=-\frac{\sqrt{6}}{7}$$

Step-by-Step Solution

Answer

Yes, \( \cot \theta = -\frac{\sqrt{6}}{7} \) is possible.

Step 1: Understanding Cotangent

Cotangent is defined as the ratio of the cosine to the sine of an angle, i.e., \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). The cotangent function can take on any real number value.

Step 2: Check for Possibility

Since the cotangent function, which is the ratio \( \frac{\cos \theta}{\sin \theta} \), can take any real number, \( \cot \theta = -\frac{\sqrt{6}}{7} \) is indeed possible. This means there must exist some angle \( \theta \) in some quadrant where the sine and cosine values satisfy this ratio.

Step 3: Determine Quadrant

Given that \( \cot \theta = -\frac{\sqrt{6}}{7} \), \( \theta \) must be in either the second or fourth quadrant. This is because in these quadrants, the signs of sine and cosine are such that their ratio, cotangent, is negative.

Key Concepts

Understanding Cotangent

Cotangent is a trigonometric function closely related to sine and cosine. When we talk about cotangent, denoted as \( \cot \theta \), we're looking at the ratio of the cosine to the sine of a given angle \( \theta \). This can be expressed mathamatically as:\[\cot \theta = \frac{\cos \theta}{\sin \theta} \]Unlike some trigonometric functions that are limited in their possible values (like sine and cosine, which range only between -1 and 1), cotangent can take on any real number. This means when you are dealing with cotangent, you're looking at an entire range of possible outputs from negative infinity to positive infinity.

This wide range of possible values for cotangent highlights its versatility.

• It can be positive, negative, zero, or undefined.
• Cotangent is undefined whenever sine is zero since division by zero is undefined in mathematics.

Interpreting Angle Quadrants

When working with trigonometric functions like cotangent, it's essential to understand how the angle \( \theta \) fits into the four quadrants of the unit circle. Each quadrant has specific characteristics:

• First Quadrant: Both sine and cosine are positive. Therefore, cotangent is also positive.
• Second Quadrant: Sine is positive, but cosine is negative, so cotangent is negative.
• Third Quadrant: Both sine and cosine are negative. Cotangent becomes positive again.
• Fourth Quadrant: Sine is negative, but cosine is positive, thus cotangent is negative.

Understanding these quadrant-specific sign rules helps us determine where a given cotangent value may be possible. For example, if \( \cot \theta = -\frac{\sqrt{6}}{7} \), the negativity of the value indicates that \( \theta \) lies either in the second or fourth quadrant.

Exploring Real Numbers in Trigonometry

Real numbers play a crucial role in trigonometry, especially when considering the range of values that functions like cotangent can take. A real number is any value that represents a quantity along a number line, encompassing both rational numbers (like fractions) and irrational numbers (like \( \sqrt{6} \)).

Trigonometric functions map angles to real numbers. For instance, since the cotangent function can assume any real number, there exist angles where \( \cot \theta \) can precisely equal \(-\frac{\sqrt{6}}{7}\). This idea showcases the continuity and vastness of trigonometric functions.

By applying real numbers:

• We can precisely describe trigonometric ratios that don't correspond to simple rational numbers.
• This range also allows us to explore solutions to more complex trigonometric equations.

Understanding real numbers in relation to trigonometric values enables deeper exploration and problem-solving flexibility.