Equivalent Trigonometric Expression For Sin(7π/6) A Comprehensive Guide

by ADMIN 72 views
Iklan Headers

Are you grappling with trigonometry and trying to figure out equivalent expressions for trigonometric functions? You've come to the right place! In this comprehensive guide, we will delve deep into the process of finding an expression equivalent to sin(7π/6). Trigonometry, at its core, is about understanding the relationships between angles and sides of triangles, and the unit circle provides a fantastic visual tool for this. This article aims not only to provide the answer but also to equip you with the knowledge to tackle similar problems with confidence. We will break down the concepts, explore different quadrants, and utilize trigonometric identities to arrive at the correct answer. So, let’s embark on this trigonometric journey together!

Understanding the Unit Circle

The unit circle is a circle with a radius of one unit centered at the origin (0, 0) in the Cartesian coordinate system. It's a fundamental tool in trigonometry because it allows us to visualize trigonometric functions for all angles. The angle, often denoted as θ (theta), is measured counterclockwise from the positive x-axis. A full rotation around the circle is 2π radians, which is equivalent to 360 degrees. Understanding the unit circle is crucial for finding equivalent expressions, as it helps us to visualize the angles and their corresponding sine, cosine, and tangent values. Let's dive deeper into how the unit circle works and its significance in solving trigonometric problems.

Key Concepts of the Unit Circle

  1. Coordinates on the Unit Circle: Any point on the unit circle can be represented by the coordinates (cos θ, sin θ), where θ is the angle formed with the positive x-axis. The x-coordinate represents the cosine of the angle (cos θ), and the y-coordinate represents the sine of the angle (sin θ). This is a fundamental concept and the cornerstone of understanding trigonometric values.
  2. Quadrants: The unit circle is divided into four quadrants, each spanning π/2 radians (90 degrees). The signs of sine and cosine vary in each quadrant:
    • Quadrant I (0 to π/2): Both sine and cosine are positive.
    • Quadrant II (π/2 to π): Sine is positive, cosine is negative.
    • Quadrant III (π to 3π/2): Both sine and cosine are negative.
    • Quadrant IV (3π/2 to 2π): Sine is negative, cosine is positive.
  3. Reference Angles: The reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. It helps in finding the trigonometric values of angles in different quadrants. For example, if you have an angle in the third quadrant, you can find its reference angle and use the reference angle to determine the sine and cosine values, considering the signs for that quadrant.
  4. Special Angles: Certain angles like 0, π/6, π/4, π/3, and π/2 (and their multiples) have well-known sine and cosine values. Memorizing these values and their corresponding points on the unit circle significantly speeds up problem-solving in trigonometry.

Significance of the Unit Circle

The unit circle provides a visual and intuitive way to understand trigonometric functions. It helps in:

  • Visualizing Angles: You can easily see where an angle lies on the circle and determine its quadrant, which helps in understanding the sign of trigonometric functions.
  • Finding Trigonometric Values: By knowing the coordinates (cos θ, sin θ) on the unit circle, you can directly find the sine and cosine values for any angle.
  • Understanding Periodic Nature: Trigonometric functions are periodic, meaning they repeat their values after a certain interval. The unit circle illustrates this periodic behavior, as after every 2π radians, the values repeat.
  • Simplifying Trigonometric Expressions: The unit circle helps in simplifying trigonometric expressions by allowing you to visualize and relate different angles and their functions.

By thoroughly understanding the unit circle, you can solve a wide range of trigonometric problems. It's not just a theoretical concept; it’s a practical tool that simplifies complex calculations and provides a deeper insight into trigonometry. Now that we've grasped the basics of the unit circle, let’s apply this knowledge to the problem at hand: finding an equivalent expression for sin(7π/6).

Analyzing sin(7π/6)

To find an equivalent expression for sin(7π/6), we first need to understand where this angle lies on the unit circle. The angle 7π/6 radians is greater than π (which is 6π/6) and less than 3π/2 (which is 9π/6). This places the angle in the third quadrant. In the third quadrant, both the sine and cosine values are negative. Remember, the sine value corresponds to the y-coordinate on the unit circle, and the cosine value corresponds to the x-coordinate. Now, let's break down the steps to determine the equivalent expression.

Step-by-Step Analysis

  1. Determine the Quadrant: As mentioned, 7π/6 lies in the third quadrant. This is crucial because it tells us that the sine value will be negative.
  2. Find the Reference Angle: The reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. To find the reference angle for 7π/6 in the third quadrant, we subtract π (which is 6π/6) from 7π/6: Reference angle = 7π/6 - π = 7π/6 - 6π/6 = π/6 The reference angle is π/6, which is a special angle that we are familiar with.
  3. Determine the Sine of the Reference Angle: We know that sin(π/6) is 1/2. This is one of the standard trigonometric values that is helpful to memorize.
  4. Apply the Quadrant Sign: Since 7π/6 is in the third quadrant, where sine is negative, we have: sin(7π/6) = -sin(π/6) = -1/2

So, we have found that sin(7π/6) = -1/2. Now, we need to look at the options provided and determine which one is also equal to -1/2. This involves analyzing each option and using our understanding of the unit circle and reference angles.

Evaluating the Options

Now that we know sin(7π/6) = -1/2, let's examine the given options and see which one matches this value. We'll go through each option step by step, using our knowledge of the unit circle and trigonometric functions.

Option A: sin(π/6)

  • Value: sin(π/6) is a standard trigonometric value. The angle π/6 (or 30 degrees) is in the first quadrant, where sine is positive. The value of sin(π/6) is 1/2.
  • Comparison: Since sin(7π/6) = -1/2 and sin(π/6) = 1/2, option A is not equivalent.

Option B: sin(5π/6)

  • Quadrant: The angle 5π/6 is in the second quadrant. In the second quadrant, sine is positive.
  • Reference Angle: To find the reference angle, we subtract 5π/6 from π (which is 6π/6): Reference angle = π - 5π/6 = 6π/6 - 5π/6 = π/6
  • Value: sin(5π/6) = sin(π/6) = 1/2 (since sine is positive in the second quadrant).
  • Comparison: Since sin(7π/6) = -1/2 and sin(5π/6) = 1/2, option B is not equivalent.

Option C: sin(5π/3)

  • Quadrant: The angle 5π/3 is in the fourth quadrant. In the fourth quadrant, sine is negative.
  • Reference Angle: To find the reference angle, we subtract 5π/3 from 2π (which is 6π/3): Reference angle = 2π - 5π/3 = 6π/3 - 5π/3 = π/3
  • Value: sin(5π/3) = -sin(π/3) = -√3/2 (since sine is negative in the fourth quadrant).
  • Comparison: Since sin(7π/6) = -1/2 and sin(5π/3) = -√3/2, option C is not equivalent.

Option D: sin(11π/6)

  • Quadrant: The angle 11π/6 is in the fourth quadrant. In the fourth quadrant, sine is negative.
  • Reference Angle: To find the reference angle, we subtract 11π/6 from 2π (which is 12π/6): Reference angle = 2π - 11π/6 = 12π/6 - 11π/6 = π/6
  • Value: sin(11π/6) = -sin(π/6) = -1/2 (since sine is negative in the fourth quadrant).
  • Comparison: Since sin(7π/6) = -1/2 and sin(11π/6) = -1/2, option D is equivalent.

After analyzing all the options, we can confidently conclude that option D, sin(11π/6), is equivalent to sin(7π/6). This meticulous evaluation demonstrates how understanding quadrants, reference angles, and trigonometric values helps in solving such problems.

Conclusion: The Equivalent Expression

In this detailed exploration, we aimed to find an expression equivalent to sin(7π/6). We began by understanding the unit circle, a fundamental tool in trigonometry, and how it helps visualize trigonometric functions and their values. We then analyzed the given expression, sin(7π/6), determining its quadrant and reference angle, which led us to find its value as -1/2. Subsequently, we evaluated each option using similar techniques, comparing their values to -1/2. Through this process, we found that:

The correct answer is D. sin(11π/6)

Key Takeaways

  • Understanding the Unit Circle: The unit circle is a powerful tool for visualizing trigonometric functions and their values. It helps in understanding the signs of sine, cosine, and tangent in different quadrants.
  • Finding Reference Angles: Reference angles simplify the process of finding trigonometric values for angles outside the first quadrant. By finding the reference angle, you can use known trigonometric values for acute angles and adjust the sign based on the quadrant.
  • Quadrant Analysis: Knowing which quadrant an angle lies in is crucial for determining the sign of its trigonometric functions. Sine is positive in the first and second quadrants, and negative in the third and fourth quadrants.
  • Step-by-Step Evaluation: Breaking down the problem into smaller steps—determining the quadrant, finding the reference angle, and applying the correct sign—makes the process more manageable and less prone to errors.

Final Thoughts

Trigonometry can seem daunting at first, but with a solid understanding of the basic concepts like the unit circle and reference angles, you can tackle a wide variety of problems. The key is to practice and apply these concepts in different scenarios. By mastering these fundamentals, you’ll be well-equipped to solve more complex trigonometric problems and gain a deeper appreciation for the beauty and utility of mathematics. Keep practicing, keep exploring, and you'll find that trigonometry becomes not just manageable but also fascinating.