Trigonometric Analysis Of Θ = 3π/4 Identifying True Statements

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In the realm of trigonometry, understanding the relationships between angles and trigonometric functions is paramount. This article delves into a specific scenario where the measure of angle θ is 3π/4 radians. We will meticulously analyze the given statements to determine their veracity, providing a comprehensive explanation for each conclusion. This exploration will not only reinforce fundamental trigonometric concepts but also illuminate the practical application of these principles in problem-solving.

Decoding the Angle θ = 3π/4

To accurately assess the statements, we must first decode the angle θ = 3π/4. This angle, expressed in radians, lies in the second quadrant of the unit circle. The unit circle, a cornerstone of trigonometry, provides a visual representation of trigonometric functions for all angles. In this circle, the angle is measured counterclockwise from the positive x-axis. To visualize 3π/4, imagine dividing the circle into four equal parts (each representing π/2 radians). The angle 3π/4 corresponds to three of these parts, placing it firmly in the second quadrant. Understanding the quadrant in which an angle resides is crucial, as it dictates the signs of the trigonometric functions associated with that angle. In the second quadrant, sine is positive, while cosine and tangent are negative. This crucial piece of information will guide our analysis of the given statements. We will also discuss how to convert radians to degrees, which will be essential for evaluating the first statement concerning the reference angle. By converting radians to degrees, and by understanding the properties of trigonometric functions in the second quadrant, we will be prepared to dissect each statement with precision and clarity. This foundation is critical for navigating the intricacies of the problem and arriving at accurate conclusions. The concepts of the unit circle and quadrant signs are fundamental building blocks in trigonometry, and mastering these concepts is key to success in more advanced topics.

A. The Measure of the Reference Angle is 45°

This statement pertains to the reference angle associated with θ = 3π/4. The reference angle, a critical concept in trigonometry, is the acute angle formed between the terminal side of the given angle and the x-axis. It simplifies the calculation of trigonometric functions for angles beyond the first quadrant. To determine the reference angle for 3π/4, we first need to understand its relationship to the nearest x-axis. Since 3π/4 lies in the second quadrant, the nearest x-axis is the negative x-axis (π radians). Therefore, the reference angle is the difference between π and 3π/4. Mathematically, this is expressed as: Reference angle = π - 3π/4 = π/4. Now, we must convert this radian measure to degrees to match the statement. The conversion factor between radians and degrees is 180°/π. Applying this, we get: Reference angle (in degrees) = (π/4) * (180°/π) = 45°. Therefore, the statement that the measure of the reference angle is 45° is TRUE. Understanding reference angles is crucial because the trigonometric function values of an angle and its reference angle have the same absolute value. The only difference lies in the sign, which is determined by the quadrant in which the original angle resides. This principle allows us to find trigonometric values for angles in any quadrant by relating them to their corresponding acute angles in the first quadrant. The ability to efficiently calculate reference angles is a fundamental skill in trigonometry and forms the basis for solving a wide range of problems.

B. tan(θ) = 1

This statement asserts that the tangent of θ = 3π/4 is equal to 1. To evaluate this, we must recall the definition of the tangent function and its behavior in the second quadrant. The tangent function, denoted as tan(θ), is defined as the ratio of the sine of the angle to its cosine: tan(θ) = sin(θ) / cos(θ). For θ = 3π/4, we need to find sin(3π/4) and cos(3π/4). As we established earlier, 3π/4 lies in the second quadrant, where sine is positive and cosine is negative. The reference angle for 3π/4 is π/4 or 45°. We know that sin(π/4) = √2/2 and cos(π/4) = √2/2. Therefore, sin(3π/4) = √2/2 (positive in the second quadrant) and cos(3π/4) = -√2/2 (negative in the second quadrant). Now, we can calculate tan(3π/4): tan(3π/4) = sin(3π/4) / cos(3π/4) = (√2/2) / (-√2/2) = -1. Thus, the statement tan(θ) = 1 is FALSE. The tangent of 3π/4 is -1, not 1. This analysis highlights the importance of considering the signs of trigonometric functions in different quadrants. A common mistake is to overlook the negative sign in the second quadrant for cosine and tangent, which can lead to incorrect conclusions. This example demonstrates how a thorough understanding of the definitions of trigonometric functions and their quadrant-specific behavior is essential for accurate problem-solving.

C. cos(θ) = -√2/2

This statement claims that the cosine of θ = 3π/4 is equal to -√2/2. As we discussed previously, 3π/4 resides in the second quadrant, where the cosine function is negative. To determine the value of cos(3π/4), we can use the reference angle, which we found to be π/4 or 45°. We know that cos(π/4) = √2/2. Since cosine is negative in the second quadrant, cos(3π/4) = -√2/2. Therefore, the statement cos(θ) = -√2/2 is TRUE. This statement reinforces the principle that the sign of a trigonometric function depends on the quadrant in which the angle lies. The ability to quickly determine the sign of a trigonometric function based on its quadrant is a valuable skill in trigonometry. It allows for efficient evaluation of trigonometric expressions and helps prevent errors in calculations. This example illustrates how understanding the reference angle concept and the quadrant signs of trigonometric functions can lead to a straightforward determination of the cosine value.

Conclusion: Correct Answers Identified

In summary, after a meticulous analysis of each statement, we have determined that:

  • Statement A (The measure of the reference angle is 45°) is TRUE.
  • Statement B (tan(θ) = 1) is FALSE.
  • Statement C (cos(θ) = -√2/2) is TRUE.

Therefore, the correct answers are A and C. This exercise underscores the importance of a strong foundation in trigonometric principles, including the unit circle, reference angles, quadrant signs, and the definitions of trigonometric functions. By mastering these concepts, one can confidently navigate and solve a wide range of trigonometric problems. This comprehensive analysis serves as a valuable resource for students and anyone seeking a deeper understanding of trigonometric functions and their applications. The ability to accurately evaluate trigonometric functions for various angles is a cornerstone of mathematics, with applications extending to physics, engineering, and other scientific disciplines. This in-depth exploration of the angle 3π/4 provides a solid framework for further learning and problem-solving in trigonometry.