Finding Sin(θ/2) Given Cos(θ) And A Quadrant

by Alex Johnson 45 views

n mathematics, especially trigonometry, we often encounter problems that require us to find the values of trigonometric functions for angles that are related to each other. One common scenario involves finding the sine, cosine, or tangent of half an angle, given the value of a trigonometric function for the full angle and the quadrant in which the angle lies. This article dives into such a problem, providing a step-by-step solution and explanation to help you understand the underlying concepts.

Understanding the Problem

The core of the problem lies in utilizing trigonometric identities, specifically the half-angle formulas, and understanding how the quadrant in which the angle lies affects the sign of the trigonometric functions. The half-angle formulas allow us to express trigonometric functions of θ/2 in terms of trigonometric functions of θ. The quadrant information is crucial because it determines whether the sine, cosine, and tangent functions are positive or negative.

Problem Statement

The problem we'll address is as follows:

If 270° < θ < 360° and cos θ = 2/3, find sin(θ/2).

We are given that θ lies in the fourth quadrant (between 270° and 360°) and the cosine of θ is 2/3. Our goal is to determine the value of sin(θ/2). This requires careful consideration of both the half-angle formula for sine and the quadrant in which θ/2 lies.

Why This Problem Matters

Problems like this are fundamental in trigonometry and have applications in various fields, including physics, engineering, and computer graphics. Mastering the techniques to solve these problems enhances your understanding of trigonometric identities and their applications. It also reinforces the importance of considering the domain and range of trigonometric functions.

Step-by-Step Solution

To solve this problem effectively, we will break it down into several steps:

  1. Determine the quadrant of θ/2.
  2. Apply the half-angle formula for sine.
  3. Substitute the given values and simplify.
  4. Consider the sign of sin(θ/2) based on the quadrant.

Step 1: Determine the Quadrant of θ/2

Knowing the quadrant in which θ/2 lies is crucial because it determines the sign of sin(θ/2). We are given that 270° < θ < 360°. To find the range for θ/2, we divide all parts of the inequality by 2:

270°/2 < θ/2 < 360°/2

135° < θ/2 < 180°

This tells us that θ/2 lies in the second quadrant. In the second quadrant, sine is positive.

Step 2: Apply the Half-Angle Formula for Sine

The half-angle formula for sine is:

sin(θ/2) = ±√[(1 - cos θ) / 2]

The ± sign indicates that we need to consider both positive and negative roots. However, since we know from Step 1 that θ/2 lies in the second quadrant where sine is positive, we will only consider the positive root.

Step 3: Substitute the Given Values and Simplify

We are given that cos θ = 2/3. Substituting this into the half-angle formula, we get:

sin(θ/2) = √[(1 - 2/3) / 2]

Now, we simplify the expression:

sin(θ/2) = √[(1/3) / 2]

sin(θ/2) = √(1/6)

sin(θ/2) = 1/√6

To rationalize the denominator, we multiply both the numerator and the denominator by √6:

sin(θ/2) = (1 * √6) / (√6 * √6)

sin(θ/2) = √6 / 6

Step 4: Consider the Sign of sin(θ/2) Based on the Quadrant

As determined in Step 1, θ/2 lies in the second quadrant, where sine is positive. Therefore, our result should be positive, which it is.

Final Answer

The value of sin(θ/2) is √6 / 6. This corresponds to option C in the given choices.

Common Mistakes and How to Avoid Them

  1. Forgetting the ± Sign in the Half-Angle Formula: One of the most common mistakes is overlooking the ± sign in the half-angle formula. Always remember to include it and then determine the correct sign based on the quadrant of the angle.
  2. Incorrectly Determining the Quadrant: Make sure to correctly determine the quadrant in which θ/2 lies. This is crucial for choosing the correct sign for the trigonometric function.
  3. Arithmetic Errors: Be careful with your arithmetic, especially when simplifying fractions and square roots. A small mistake can lead to an incorrect answer.
  4. Not Rationalizing the Denominator: While not always required, it is good practice to rationalize the denominator to present the answer in its simplest form.

Practice Problems

To solidify your understanding, try solving these similar problems:

  1. If 180° < θ < 270° and cos θ = -1/4, find sin(θ/2).
  2. If 90° < θ < 180° and sin θ = 3/5, find cos(θ/2).
  3. If 0° < θ < 90° and tan θ = 1, find tan(θ/2).

Working through these problems will help you become more comfortable with applying half-angle formulas and considering the quadrant of the angle.

Conclusion

Finding sin(θ/2) given cos θ and the quadrant of θ involves a systematic approach using the half-angle formula for sine and considering the quadrant in which θ/2 lies. By carefully following the steps outlined in this article, you can solve these problems accurately and confidently. Remember to pay attention to the sign of the trigonometric function based on the quadrant and avoid common mistakes. Trigonometry is a fascinating branch of mathematics with numerous applications, and mastering these fundamental concepts will serve you well in more advanced studies.

For further exploration of trigonometric identities and half-angle formulas, you can visit trusted websites like Khan Academy's Trigonometry Section.