Angle 11π/6 Trigonometric Analysis

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In the realm of mathematics, trigonometry stands as a pillar of understanding angles and their relationships with the sides of triangles. A cornerstone of this field involves deciphering angles expressed in radians and their corresponding trigonometric values. This article dives deep into the intricacies of the angle θ = 11π/6, meticulously examining its reference angle, tangent, and cosine to determine the veracity of several statements. We will embark on a detailed exploration to dissect each option, offering clarity and a comprehensive understanding. Whether you're a student grappling with trigonometry or a math aficionado seeking to reinforce your knowledge, this guide provides a thorough analysis of the angle θ = 11π/6.

Understanding the Angle θ = 11π/6

To truly understand the angle θ = 11π/6, we must first grasp the concept of radians. Radians are a unit of angular measure, much like degrees, but are based on the radius of a circle. One radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. A full circle encompasses 2π radians, which corresponds to 360 degrees. Therefore, π radians equate to 180 degrees. To convert radians to degrees, we use the formula: degrees = radians * (180/π). Applying this to our angle, θ = (11π/6) * (180/π) = 330 degrees. This conversion places our angle in the fourth quadrant of the unit circle, where the x-coordinate (cosine) is positive and the y-coordinate (sine) is negative.

Reference Angle

The reference angle is the acute angle formed by the terminal side of the given angle and the x-axis. It's crucial for simplifying trigonometric calculations, as it provides a standard angle within the first quadrant (0 to 90 degrees or 0 to π/2 radians) to which we can relate our given angle. To find the reference angle for θ = 330 degrees, we subtract it from 360 degrees: 360 - 330 = 30 degrees. Alternatively, in radians, the reference angle is 2π - (11π/6) = π/6 radians. Converting π/6 radians to degrees, we get (π/6) * (180/π) = 30 degrees. Therefore, statement A, which claims the reference angle is 60 degrees, is incorrect. The correct reference angle is 30 degrees, which is a fundamental value in trigonometry, corresponding to special triangles that we can leverage to find trigonometric ratios.

Tangent of θ

The tangent of an angle, denoted as tan(θ), is a fundamental trigonometric function defined as the ratio of the sine of the angle to its cosine (tan(θ) = sin(θ) / cos(θ)). In the context of the unit circle, tan(θ) can also be visualized as the slope of the line that forms the angle θ with the positive x-axis. To determine the tangent of θ = 11π/6, we first need to find the sine and cosine of this angle. As we established, θ = 11π/6 lies in the fourth quadrant, where sine is negative and cosine is positive. The reference angle is π/6 (or 30 degrees), which helps us find the magnitudes of sine and cosine. The sine of π/6 is 1/2, and the cosine of π/6 is √3/2. Therefore, sin(11π/6) = -1/2 (since sine is negative in the fourth quadrant), and cos(11π/6) = √3/2.

Now we can calculate tan(11π/6) = sin(11π/6) / cos(11π/6) = (-1/2) / (√3/2) = -1/√3. To rationalize the denominator, we multiply both the numerator and denominator by √3, which gives us tan(11π/6) = -√3/3. This value is approximately -0.577, clearly indicating that tan(11π/6) is not equal to 1. Consequently, statement B, which asserts that tan(θ) = 1, is definitively false. The tangent of the angle is a negative value, a crucial detail that aligns with the angle's position in the fourth quadrant. Understanding the sign conventions in different quadrants is essential for accurate trigonometric calculations.

Cosine of θ

The cosine of an angle, often abbreviated as cos(θ), is a critical trigonometric function that relates an angle to the ratio of the adjacent side to the hypotenuse in a right-angled triangle. In the context of the unit circle, cos(θ) corresponds to the x-coordinate of the point where the terminal side of the angle intersects the circle. For the angle θ = 11π/6, we previously established that it lies in the fourth quadrant, where cosine is positive. The reference angle for θ = 11π/6 is π/6 (or 30 degrees). The cosine of the reference angle, cos(π/6), is a well-known trigonometric value equal to √3/2.

Since cosine is positive in the fourth quadrant, the cosine of θ = 11π/6 is also √3/2. Therefore, statement C, which claims that cos(θ) = √3/2, is indeed true. The cosine value provides a direct measure of the x-coordinate on the unit circle, offering insight into the angle's horizontal position. This calculation underscores the importance of recognizing the quadrant in which the angle lies, as this determines the sign of the trigonometric function. The precise value of cos(θ) = √3/2 is a testament to the predictable and consistent nature of trigonometric functions within the unit circle framework.

In conclusion, after a thorough examination of the statements related to the angle θ = 11π/6, we have determined that only statement C, asserting that cos(θ) = √3/2, is true. Statement A incorrectly identified the reference angle as 60 degrees when it is, in fact, 30 degrees. Statement B falsely claimed that tan(θ) = 1, whereas the correct value is -√3/3. This exercise underscores the importance of understanding reference angles, quadrant rules, and the definitions of trigonometric functions in accurately evaluating trigonometric expressions. Mastering these concepts is essential for success in trigonometry and its applications in various fields, including physics, engineering, and computer graphics. The angle θ = 11π/6 serves as a valuable case study for reinforcing these fundamental principles and enhancing problem-solving skills in trigonometry.

Decoding Angle 11π/6 Trigonometric Analysis and Solutions