Problem 134 - A Graphical Approach to Precalculus with Limits

Question

Work each problem. CONCEPT CHECK True or false? Since cot \(\theta=\frac{\cos \theta}{\sin \theta}\) if cot \(\theta=\frac{1}{2}\) with \(\theta\) in quadrant \(I,\) then \(\cos \theta=1\) and \(\sin \theta=2 .\) If false, explain why.

Step-by-Step Solution

Answer

False; \\( \\sin \\theta \\leq 1 \\,\\ therefore \\sin \\theta \\\neq 2 \\) and violates the Pythagorean Identity.

Step 1: Understanding the Expression

The cotangent function is defined as \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). We are given that \( \cot \theta = \frac{1}{2} \) and that \( \theta \) is in Quadrant I.

Step 2: Evaluate \\( \\\frac{\\\\cos \\theta}{\\\\sin \\theta} \\) with the Given \\\( cot \\theta \\\)

Since \( \cot \theta = \frac{1}{2} \), it follows that \( \frac{\cos \theta}{\sin \theta} = \frac{1}{2} \). This implies \( \cos \theta = \frac{1}{2} \cdot \sin \theta \).

Step 3: Check the Proposition \( \cos \theta = 1 \), \( \sin \theta = 2 \)

For \( \cos \theta = 1 \) and \( \sin \theta = 2 \), the expression \( \frac{\cos \theta}{\sin \theta} = \frac{1}{2} \) becomes \( \frac{1}{2} = \frac{1}{2} \). However, this assignment results in \( \sin \theta = 2 \), which is impossible based on the Pythagorean Identity.

Step 4: Apply the Pythagorean Identity Constraint

According to the Pythagorean Identity \( \sin^2 \theta + \cos^2 \theta = 1 \), if \( \sin \theta = 2 \), then \( 2^2 + \cos^2 \theta = 1 \), which leads to a negative value for \( \cos^2 \theta \), thus invalidating the proposed values.

Step 5: Conclusion Assessment

Given that \( \sin \theta \) cannot be 2 as it violates the Pythagorean Identity, the statement \( \cos \theta = 1 \) and \( \sin \theta = 2 \) is false.

Key Concepts

Cotangent Function

The cotangent function, abbreviated as \( \cot \theta \), is a fundamental trigonometric function often introduced in the study of right-angled triangles. It is defined as the ratio of the cosine of an angle to the sine of that angle. Mathematically, this can be expressed as:

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

This definition reveals that the cotangent is essentially the reciprocal of the tangent function, which is the ratio of sine to cosine. Understanding this function is crucial because it provides insights into the relationships between the trigonometric functions beyond the familiar sine and cosine.

In trigonometric identities and equations, knowing about cotangent helps simplify expressions and solve problems, especially when working with angles in specific quadrants or requiring transformation of one trigonometric function into another. In this particular exercise, \( \cot \theta \) is given as \( \frac{1}{2} \). This tells us the relationship between \( \sin \theta \) and \( \cos \theta \) without needing their exact values at first glance.

By understanding cotangent, students can also grasp important reciprocal relationships, aiding further explorations in trigonometry.

Pythagorean Identity

The Pythagorean Identity is one of the most critical identities in trigonometry. It states:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity is based on the Pythagorean theorem, applied within the unit circle framework. It underscores that no matter what angle \( \theta \) you have, the sum of the squares of sine and cosine values will always equal one.

In the context of this exercise, the Pythagorean Identity is used to challenge the proposed values of \( \cos \theta = 1 \) and \( \sin \theta = 2 \). Plugging these values into the identity would yield \( 1^2 + 2^2 = 1 \), or \( 1 + 4 = 1 \), which is clearly incorrect. This reveals that one or both of the initial assumptions about \( \cos \theta \) and \( \sin \theta \) were incorrect.

Remembering and utilizing the Pythagorean Identity is vital for verifying the plausibility of trigonometric values, ensuring that the derived solutions are valid within the realm of trigonometry. It acts as a reliable tool to confirm the consistency and correctness of trigonometric equations.

Quadrant I

Trigonometry divides the coordinate plane into four quadrants. Each quadrant has unique characteristics based on the signs of trigonometric functions. Quadrant I, where \( 0^\circ < \theta < 90^\circ \), is the home for positive values of both sine and cosine. As such, all primary trigonometric ratios (sine, cosine, tangent, cosecant, secant, and cotangent) are positive in this quadrant.

In the problem given, \( \theta \) is specified to be in Quadrant I, informing us that both \( \sin \theta \) and \( \cos \theta \) should only yield positive results. Understanding this quadrant system helps eliminate impossible value assignments, like \( \sin \theta = 2 \), because sine and cosine values must be less than or equal to 1, as per the definition of sine and cosine in the unit circle.

This quadrant context provides a foundation for solving and constraining trigonometric problems, ensuring that solutions remain within feasible and correct boundaries. Students should always verify the quadrant range of an angle before finalizing trigonometric value assignments.