Problem 32 - Precalculus

Question

Find the indicated trigonometric function values. If \(\cot \theta=1,\) and the terminal side of \(\theta\) lies in quadrant \(I,\) find \(\sin \theta\)

Step-by-Step Solution

Answer

\( \sin \theta = \frac{\sqrt{2}}{2} \)

Step 1: Understand the given information

We are given that the cotangent of an angle \( \theta \) is \( \cot \theta = 1 \) and that \( \theta \) lies in the first quadrant. Our goal is to find \( \sin \theta \).

Step 2: Use cotangent definition

Recall that \( \cot \theta \) is the reciprocal of \( \tan \theta \), so \( \cot \theta = \frac{1}{\tan \theta} \). Since \( \cot \theta = 1 \), we have \( \tan \theta = 1 \).

Step 3: Find the reference angle

The equation \( \tan \theta = 1 \) reminds us that in the unit circle this occurs at \( \theta = \frac{\pi}{4} \) radians or \( 45^\circ \) since both sine and cosine of this angle are equal, making the ratio \( \tan \theta = \frac{1}{1} = 1 \).

Step 4: Determine sine of the angle

Since \( \theta = \frac{\pi}{4} \) (or \( 45^\circ \)) in the first quadrant, we know that \( \sin \theta = \frac{\sqrt{2}}{2} \) because for angle \( \frac{\pi}{4} \), both sine and cosine are equal to \( \frac{\sqrt{2}}{2} \).

Step 5: Verify first quadrant conditions

In the first quadrant, all trigonometric functions are positive. Since \( \sin \theta = \frac{\sqrt{2}}{2} \) is positive, it confirms that we have the correct value given the condition.

Key Concepts

Quadrant System

The quadrant system is an essential part of understanding trigonometry. It divides the coordinate plane into four distinct sections. Quadrants are designated as I, II, III, and IV, moving counterclockwise from the positive x-axis. Let's delve into some key characteristics of each quadrant:

• Quadrant I: Both x and y coordinates are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: Both x and y coordinates are negative.
• Quadrant IV: x is positive, y is negative.

In trigonometry, the behavior of the trigonometric functions can vary depending on the quadrant. For example, in Quadrant I, all six trigonometric functions are positive. As a result, when we are given that an angle lies in Quadrant I, it guides us in determining the signs of trigonometric values like sine and cotangent. Understanding which quadrant an angle sits in allows us to solve problems quicker by anticipating these signs.

Sine Function

The sine function is a fundamental trigonometric function that associates an angle with the ratio of the opposite side to the hypotenuse in a right-angled triangle. This is often expressed as:\[\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\]In the context of the unit circle, where the radius is 1, the sine of an angle represents the y-coordinate of a point where the terminal side of the angle intersects the circle.
The sine function varies smoothly between -1 and 1. It is important to note that sine values are positive in Quadrants I and II because the y-coordinates are positive in these quadrants.

• In Quadrant I: \( \sin \theta > 0 \)
• In Quadrant II: \( \sin \theta > 0 \)
• In Quadrant III: \( \sin \theta < 0 \)
• In Quadrant IV: \( \sin \theta < 0 \)

In our problem, we determined that the sine of an angle, \( \theta = \frac{\pi}{4} \), in the first quadrant is \( \frac{\sqrt{2}}{2} \). This ensures that the sine function aligns with the positive nature expected in Quadrant I.

Cotangent Function

The cotangent function is another important trigonometric function. It is defined as the reciprocal of the tangent function. Mathematically, it is represented as:\[\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}\]In a right triangle, it describes the ratio of the length of the adjacent side to the opposite side. Since \( \cot \theta \) is the inverse of \( \tan \theta \), its properties are directly influenced by that of tangent.

• Positive in Quadrants I and III: Where both tangent and cotangent have positive values.
• Negative in Quadrants II and IV: Reflecting tangents negative values.

In this specific problem, knowing \( \cot \theta = 1 \) leads us to conclude that \( \tan \theta \) must also be 1. This situation occurs naturally when angle \( \theta = \frac{\pi}{4} \) or \( 45^\circ \), as both sine and cosine of such angles equal \( \frac{\sqrt{2}}{2} \). Understanding cotangent's reciprocal nature helps us solve similar exercises, supplementing the process of determining other function values.