Cos(10π/3): A Step-by-Step Solution

Have you ever stumbled upon a seemingly complex trigonometric problem and felt a bit lost? Don't worry, we've all been there! Trigonometry can appear daunting at first, but breaking it down into manageable steps can make it surprisingly straightforward. Today, we're going to walk through a classic example: finding the exact value of cos(10π/3). We'll follow Henry's method, which is a fantastic way to approach these types of problems. So, let's put on our math hats and dive in!

1. Simplifying the Angle: Finding the Coterminal Angle

In this initial step, the main goal is to simplify the angle (10π/3) to something more manageable. Think of it like this: angles that differ by multiples of 2π (a full circle) are coterminal, meaning they end up at the same spot on the unit circle. And because trigonometric functions are periodic, coterminal angles have the same trigonometric values. Henry's approach of subtracting 2π is exactly what we need to do.

The question at hand, finding the exact value of cos(10π/3), can seem tricky due to the size of the angle. A large angle like 10π/3 makes it difficult to visualize its position on the unit circle directly. This is where the concept of coterminal angles becomes incredibly useful. Coterminal angles are angles that share the same initial and terminal sides. In simpler terms, they are angles that, when drawn in standard position, end up pointing in the same direction. The beauty of coterminal angles lies in the fact that trigonometric functions (sine, cosine, tangent, etc.) have the same values for coterminal angles. This is because these functions depend on the position of the terminal side, not the number of rotations taken to get there.

To find a coterminal angle, we can add or subtract multiples of 2π (which represents a full rotation) from the given angle. The idea is to bring the angle within the range of 0 to 2π (or -2π to 0), which corresponds to one full revolution around the unit circle. This simplified angle is much easier to work with and visualize. In Henry's case, he correctly recognizes that subtracting 2π from 10π/3 will give him a coterminal angle that is easier to handle. This is a crucial first step in solving the problem, as it transforms a potentially unwieldy angle into one that is readily identifiable on the unit circle.

So, when you encounter a trigonometric problem with a large or seemingly complex angle, always consider finding a coterminal angle first. This simple step can significantly simplify the problem and make it much easier to solve. By subtracting multiples of 2π, you're essentially stripping away the full rotations and focusing on the essential part of the angle that determines its trigonometric values. This technique is a cornerstone of trigonometry and a must-have in your problem-solving toolkit. Understanding coterminal angles is not just about simplifying calculations; it's about gaining a deeper understanding of the periodic nature of trigonometric functions and how angles relate to positions on the unit circle.

Let's look at the math:

10π/3 - 2π = 10π/3 - 6π/3 = 4π/3

So, cos(10π/3) = cos(4π/3). Much better, right? 4π/3 is an angle we can easily visualize on the unit circle. Now, let's see what's next.

2. Visualizing the Unit Circle: Locating 4π/3

Now that we've simplified our angle to 4π/3, the next step is to visualize it on the unit circle. The unit circle is your best friend in trigonometry! It's a circle with a radius of 1, centered at the origin of a coordinate plane. Angles are measured counterclockwise from the positive x-axis. To truly grasp the essence of trigonometry, you need to be comfortable navigating the unit circle. It’s not just a geometric figure; it’s a powerful tool that connects angles to their trigonometric values in a visually intuitive way. The unit circle serves as a map, guiding us through the landscape of trigonometric functions and their relationships. Understanding how angles correspond to points on the unit circle is fundamental to solving trigonometric problems. It allows us to move beyond rote memorization and develop a deeper, more conceptual understanding.

When we talk about the unit circle, we're essentially talking about a circle with a radius of 1, centered at the origin (0, 0) of a coordinate plane. The beauty of this circle lies in its simplicity and the direct relationship it establishes between angles and their trigonometric values. Each point on the unit circle can be represented by coordinates (x, y), where x represents the cosine of the angle and y represents the sine of the angle. This connection is the heart of the unit circle's power.

Angles are measured counterclockwise from the positive x-axis. A full rotation around the circle is 2π radians (or 360 degrees). Key angles like 0, π/2, π, and 3π/2 radians (or 0, 90, 180, and 270 degrees) mark the cardinal directions on the circle. To locate an angle like 4π/3 on the unit circle, we need to understand how radians correspond to fractions of a full circle. Since 2π radians represent a full circle, π radians represent half a circle. Therefore, 4π/3 radians represent more than half a circle but less than a full circle. Specifically, it's π/3 radians (or 60 degrees) past π radians (180 degrees). This places the angle 4π/3 in the third quadrant of the unit circle.

Visualizing angles on the unit circle is not just about memorizing positions; it's about developing a spatial understanding of angles and their relationships. It's about seeing how angles relate to each other, how they transform under rotations, and how they project onto the x and y axes. This visual intuition is invaluable when solving trigonometric problems. For instance, understanding the symmetry of the unit circle allows us to quickly determine the trigonometric values of related angles. By mastering the unit circle, you're not just learning trigonometry; you're developing a powerful visual tool that will serve you well in many areas of mathematics and beyond.

So, 4π/3 is in the third quadrant. Remember that the quadrants are numbered counterclockwise, starting from the top right. This is important because it tells us about the signs of the cosine and sine values in that quadrant. In the third quadrant, both x (cosine) and y (sine) are negative. This knowledge will help us determine the sign of our final answer. Think of it like a treasure map – the unit circle guides us, and the quadrant clues us in on the final find!

3. Reference Angles: Finding the Related Acute Angle

Once we've located 4π/3, we need to find its reference angle. The reference angle is the acute angle (less than π/2 or 90 degrees) formed between the terminal side of our angle and the x-axis. It’s like finding the “shortest distance” back to the x-axis. Why do we need this? Because reference angles help us relate trigonometric values of angles in different quadrants to values we already know from the first quadrant. Mastering the concept of reference angles is like unlocking a secret code in trigonometry. It allows us to leverage our knowledge of trigonometric values in the first quadrant to determine the values in all other quadrants. This significantly simplifies calculations and enhances our understanding of trigonometric functions.

The reference angle is essentially the acute angle formed between the terminal side of our angle and the x-axis. It's always a positive angle, less than 90 degrees (or π/2 radians). Think of it as the “shadow” of our angle in the first quadrant. To find the reference angle, we need to consider which quadrant our angle lies in. For angles in the first quadrant, the reference angle is simply the angle itself. However, for angles in other quadrants, we need to perform a simple calculation. For angles in the second quadrant, we subtract the angle from π (or 180 degrees). For angles in the third quadrant, we subtract π (or 180 degrees) from the angle. And for angles in the fourth quadrant, we subtract the angle from 2π (or 360 degrees).

The power of reference angles lies in their ability to connect trigonometric values across quadrants. Trigonometric functions have the same absolute values for an angle and its reference angle. The only difference is the sign (positive or negative), which depends on the quadrant in which the original angle lies. This is because the reference angle essentially “folds” the angle back into the first quadrant, where we have a clear understanding of trigonometric values. By understanding the signs of trigonometric functions in each quadrant (using mnemonics like “All Students Take Calculus”), we can easily determine the correct sign for our answer.

The use of reference angles allows us to simplify trigonometric calculations significantly. Instead of directly calculating the trigonometric values of a large or complex angle, we can find its reference angle, determine the corresponding trigonometric value in the first quadrant, and then adjust the sign based on the original angle's quadrant. This approach is not only efficient but also enhances our conceptual understanding of trigonometric functions and their behavior across the unit circle. Furthermore, reference angles help us to see the symmetry and patterns within the unit circle, making trigonometry more intuitive and less reliant on rote memorization.

For 4π/3, which is in the third quadrant, the reference angle is:

4π/3 - π = π/3

So, our reference angle is π/3 (60 degrees). This is a familiar angle, and we know its trigonometric values! Remember, knowing the reference angle is like having a key that unlocks the trigonometric values. It simplifies the process and makes the calculations much easier.

4. Determining the Trigonometric Value: cos(π/3)

Now we need to find the cosine of our reference angle, π/3. This is where those special triangles (30-60-90 triangles) or the unit circle chart come in handy. Do you remember the values for common angles? Cos(π/3) is a common trigonometric value that's worth memorizing, or at least knowing how to derive quickly. Understanding the trigonometric values of special angles like π/6 (30 degrees), π/4 (45 degrees), and π/3 (60 degrees) is a cornerstone of trigonometry. These angles appear frequently in problems, and knowing their sine, cosine, and tangent values can significantly speed up your problem-solving process. There are several ways to approach memorizing these values, but one of the most effective is to understand their derivation from special triangles and the unit circle.

The most common approach to deriving these values is through the use of two special right triangles: the 30-60-90 triangle and the 45-45-90 triangle. The 30-60-90 triangle is formed by bisecting an equilateral triangle. If we assume the side length of the equilateral triangle is 2, then the sides of the 30-60-90 triangle will be 1 (opposite the 30-degree angle), √3 (opposite the 60-degree angle), and 2 (the hypotenuse). Using the definitions of sine, cosine, and tangent (SOH CAH TOA), we can easily calculate the trigonometric values for 30 degrees (π/6 radians) and 60 degrees (π/3 radians).

The 45-45-90 triangle is formed by bisecting a square along its diagonal. If we assume the side length of the square is 1, then the sides of the 45-45-90 triangle will be 1, 1, and √2 (the hypotenuse). Again, using SOH CAH TOA, we can calculate the trigonometric values for 45 degrees (π/4 radians). Another powerful way to visualize and understand these values is through the unit circle. As we discussed earlier, the unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. The coordinates of any point on the unit circle can be represented as (cos θ, sin θ), where θ is the angle formed between the positive x-axis and the line connecting the origin to the point. By drawing lines corresponding to the special angles (π/6, π/4, π/3) on the unit circle, we can visually determine the sine and cosine values as the y and x coordinates of the intersection points. This not only helps in memorization but also reinforces the connection between angles and their trigonometric values.

It's worth noting that memorizing the trigonometric values of these special angles is not just about rote learning; it's about developing a strong foundation in trigonometry. These values serve as building blocks for understanding more complex trigonometric concepts and solving a wide range of problems. With practice, these values become second nature, allowing you to tackle trigonometric challenges with greater confidence and efficiency. Remember, the goal is not just to recall the values but to understand their origins and how they relate to each other and to the unit circle.

cos(π/3) = 1/2

Great! We're almost there. Now we just need to consider the sign.

5. Determining the Sign: Quadrant Matters

Remember, 4π/3 is in the third quadrant, where cosine is negative. This is a crucial step! We've found the magnitude of the cosine value using the reference angle, but the sign depends on the quadrant. This step is all about putting the pieces together – we've got the reference angle, we know the cosine value for that reference angle, and now we need to ensure our answer has the correct sign based on the quadrant of the original angle. Mastering the sign conventions for trigonometric functions in different quadrants is a critical skill in trigonometry. It allows us to accurately determine the sign of our answer and avoid common errors. The unit circle serves as a visual guide, but understanding the underlying principles is key to long-term retention and application.

There are several mnemonics that can help you remember the sign conventions. One of the most popular is “All Students Take Calculus” (ASTC). This mnemonic tells us which trigonometric functions are positive in each quadrant. In the first quadrant (All), all trigonometric functions are positive. In the second quadrant (Students), sine and its reciprocal, cosecant, are positive. In the third quadrant (Take), tangent and its reciprocal, cotangent, are positive. And in the fourth quadrant (Calculus), cosine and its reciprocal, secant, are positive. Another helpful way to visualize this is by looking at the unit circle itself. In the first quadrant, both x and y coordinates are positive, so sine (y) and cosine (x) are positive. In the second quadrant, x is negative and y is positive, so only sine is positive. In the third quadrant, both x and y are negative, so only tangent (which is y/x) is positive. And in the fourth quadrant, x is positive and y is negative, so only cosine is positive.

Understanding why these sign conventions exist is just as important as memorizing them. The signs of trigonometric functions are directly related to the signs of the x and y coordinates on the unit circle. Sine corresponds to the y-coordinate, cosine corresponds to the x-coordinate, and tangent is the ratio of y to x. By understanding this fundamental connection, we can intuitively determine the sign of a trigonometric function in any quadrant. This approach not only makes memorization easier but also deepens our understanding of trigonometric functions and their behavior across the unit circle.

Furthermore, when using reference angles, it's essential to remember that the reference angle only gives us the magnitude (absolute value) of the trigonometric function. We still need to determine the sign based on the quadrant of the original angle. This is a common point of confusion for students, so it's worth emphasizing the distinction between the reference angle and the actual angle. By carefully considering the quadrant, we can ensure that our final answer has the correct sign. In practical problem-solving, this step is often the difference between a correct answer and an incorrect one. It's a crucial detail that demonstrates a thorough understanding of trigonometric principles.

Therefore:

cos(4π/3) = -1/2

6. The Final Answer

Putting it all together:

cos(10π/3) = cos(4π/3) = -1/2

And there you have it! We've successfully found the exact value of cos(10π/3) by breaking it down into manageable steps. Henry's method is a great example of how to approach trigonometric problems: simplify the angle, visualize it on the unit circle, find the reference angle, determine the trigonometric value, and finally, consider the sign. Remember, practice makes perfect. The more you work with trigonometric functions and the unit circle, the more comfortable you'll become. Keep practicing, and soon you'll be solving these problems with ease!

Key Takeaways

• Coterminal Angles: Simplifying angles by adding or subtracting multiples of 2π.
• Unit Circle: Visualizing angles and their trigonometric values.
• Reference Angles: Finding the related acute angle to simplify calculations.
• Quadrant Signs: Determining the correct sign based on the quadrant.

By understanding these core concepts, you'll be well-equipped to tackle a wide range of trigonometric problems. Keep exploring, keep learning, and most importantly, have fun with math!