Solving Trigonometry Problems Unveiling The Truth About Θ = 11π/6

Hey guys! Today, we're diving deep into the fascinating world of trigonometry, specifically tackling the angle θ = 11π/6. This angle might seem a bit intimidating at first glance, but don't worry, we're going to break it down step by step. Our mission? To identify which statements about this angle are true. We'll explore its reference angle, tangent, and sine values, ensuring you have a solid understanding of the concepts involved. So, buckle up and let's embark on this trigonometric adventure together!

When it comes to angles like θ = 11π/6, understanding the reference angle is crucial. The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. It helps us to easily determine the trigonometric values of angles in any quadrant. For θ = 11π/6, which lies in the fourth quadrant, we need to calculate the reference angle. The full circle is 2π, so we subtract 11π/6 from 2π to find the reference angle. This calculation gives us 2π - 11π/6 = π/6. Now, π/6 radians is equivalent to 30 degrees. Therefore, the reference angle for θ = 11π/6 is 30 degrees, not 60 degrees as stated in option A. Remembering this fundamental concept of reference angles will aid in correctly assessing trigonometric functions for any given angle. This is a cornerstone in trigonometry and helps simplify calculations and understanding of trigonometric values in various quadrants.

Moving on to the tangent of θ, tan(θ), we need to evaluate tan(11π/6). Tangent is defined as the ratio of sine to cosine, tan(θ) = sin(θ)/cos(θ). For 11π/6, we know it lies in the fourth quadrant where sine is negative and cosine is positive. The reference angle is π/6, so we can use the known trigonometric values for π/6 to find those for 11π/6. We have sin(π/6) = 1/2 and cos(π/6) = √3/2. Therefore, sin(11π/6) = -1/2 (since sine is negative in the fourth quadrant) and cos(11π/6) = √3/2. Now, tan(11π/6) = sin(11π/6) / cos(11π/6) = (-1/2) / (√3/2) = -1/√3. Rationalizing the denominator, we get -√3/3, which is definitely not equal to 1. So, statement B, stating that tan(θ) = 1, is incorrect. This step-by-step evaluation highlights the importance of understanding quadrant signs and using reference angles to accurately calculate trigonometric values.

Lastly, let's investigate the sine of θ, sin(θ). As we already determined while calculating the tangent, sin(11π/6) = -1/2. This is because in the fourth quadrant, the sine function is negative. The reference angle, π/6, has a sine of 1/2, but due to the quadrant, we take the negative value. Therefore, sin(11π/6) is indeed equal to -1/2, making statement C a true statement. This reinforces the crucial role of quadrant awareness when dealing with trigonometric functions. The sign of the trigonometric function changes depending on the quadrant in which the angle lies. Understanding this concept is key to accurately determining trigonometric values.

Let's delve a bit deeper into the concepts we've touched upon. Understanding trigonometric functions like sine, cosine, and tangent is paramount in solving such problems. These functions relate the angles of a triangle to the ratios of its sides. The unit circle provides a visual representation of these functions, making it easier to understand their behavior across different quadrants. The sine function corresponds to the y-coordinate, the cosine function to the x-coordinate, and the tangent function is the ratio of y to x. Grasping these relationships on the unit circle simplifies the calculation of trigonometric values for any angle. Furthermore, the concept of reference angles acts as a bridge between angles in different quadrants. By finding the reference angle, which is always acute, we can use known trigonometric values of angles in the first quadrant to find values in other quadrants. The sign adjustments are then made based on the quadrant the original angle lies in. This technique is invaluable in simplifying complex trigonometric problems.

To summarize our approach, we followed a step-by-step method to verify each statement. First, we focused on finding the reference angle for θ = 11π/6. This involved understanding that 11π/6 lies in the fourth quadrant and calculating the acute angle formed with the x-axis. This foundational step allowed us to relate the trigonometric values of 11π/6 to the known values of its reference angle. Next, we evaluated tan(θ), using the relationship tan(θ) = sin(θ)/cos(θ). We determined the signs of sine and cosine in the fourth quadrant and used the sine and cosine values of the reference angle to calculate tan(11π/6). The rationalization of the denominator was an important step to present the final answer in its simplest form. Finally, we directly evaluated sin(θ) by considering its sign in the fourth quadrant and the sine value of the reference angle. This systematic approach ensures accuracy and helps to avoid common pitfalls in trigonometric calculations.

It's always good to be aware of potential pitfalls, right? When working with trigonometric functions, especially in scenarios like the one we've explored, there are several common mistakes that students often make. One frequent error is neglecting to consider the quadrant in which the angle lies. As we've seen, the signs of sine, cosine, and tangent vary across quadrants, and failing to account for this can lead to incorrect answers. Another common mistake is confusion with reference angles. It's crucial to correctly identify the reference angle and relate it to the given angle. Also, errors can arise when calculating trigonometric values of special angles like π/6, π/4, and π/3. Having a solid understanding of these values and their relationships is key. Lastly, watch out for algebraic errors during calculations, especially when dealing with fractions and square roots. Rationalizing denominators, for example, requires careful attention to detail. By being mindful of these potential pitfalls, you can significantly improve your accuracy in solving trigonometric problems.

Alright guys, we've reached the end of our trigonometric journey for today! We've successfully navigated the complexities of θ = 11π/6, and hopefully, you now have a clearer understanding of reference angles, sine, and tangent. Remember, practice makes perfect! The more you work with these concepts, the more comfortable and confident you'll become. So, keep exploring, keep questioning, and keep those trigonometric gears turning. In summary, we found that statement C, which stated that sin(θ) = -1/2, is the only true statement. Statements A and B were incorrect due to miscalculations of the reference angle and the tangent value. Understanding these fundamental concepts will set you up for success in more advanced mathematical adventures. Until next time, keep those angles sharp!

• Trigonometry
• Reference Angle
• Unit Circle
• Sine
• Cosine
• Tangent
• Radians
• 11π/6
• Trigonometric Functions
• Quadrant