Solve Cos⁻¹(cos(7π/6)) Easily: Step-by-Step Guide

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Hey guys! Let's break down how to solve the expression cos⁻¹(cos(7π/6)). This might seem tricky at first, but with a step-by-step approach, it becomes super manageable. We'll go through the unit circle, reference angles, and the range of the inverse cosine function. By the end of this guide, you’ll be solving these problems like a pro! So, grab your thinking caps and let's dive in!

Understanding the Basics

Before we jump into the problem, let's refresh some key concepts. The cosine function gives us the x-coordinate of a point on the unit circle. The inverse cosine function, denoted as cos⁻¹(x) or arccos(x), asks the question: "What angle has a cosine of x?" It's crucial to remember that the range of cos⁻¹(x) is [0, π], meaning the answer must be an angle between 0 and π radians (or 0° and 180°).

Why is the range of the inverse cosine function so important? Well, the cosine function is periodic, meaning it repeats its values. For example, cos(π/3) = 1/2, but so does cos(5π/3) = 1/2. To make the inverse cosine function well-defined, we restrict its range to [0, π]. This ensures that for every value of x in the domain of cos⁻¹(x) (which is [-1, 1]), there is only one output angle.

Think of it like this: if someone asks you, "What number, when squared, equals 4?", you could say 2 or -2. But if we define the square root function to only return the positive root, we get a unique answer. Similarly, restricting the range of cos⁻¹(x) gives us a unique angle.

Now, let's consider the unit circle. The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the Cartesian plane. Angles are measured counterclockwise from the positive x-axis. For any angle θ, the coordinates of the point where the terminal side of the angle intersects the unit circle are (cos θ, sin θ). This visual representation is extremely helpful for understanding trigonometric functions and their values at various angles.

Understanding reference angles is also essential. A reference angle is the acute angle formed between the terminal side of an angle and the x-axis. For example, if we have an angle of 7π/6, it lies in the third quadrant. The reference angle is the angle between 7π/6 and the negative x-axis (π). So, the reference angle is 7π/6 - π = π/6. Reference angles help us determine the values of trigonometric functions for angles outside the range of 0 to π/2. By knowing the reference angle and the quadrant in which the angle lies, we can determine the sign and value of the trigonometric function.

Step-by-Step Solution

Okay, let's tackle cos⁻¹(cos(7π/6)) step by step.

Step 1: Evaluate the Inner Cosine Function

First, we need to find the value of cos(7π/6). As we discussed earlier, 7π/6 lies in the third quadrant. The reference angle for 7π/6 is π/6. In the third quadrant, cosine is negative. We know that cos(π/6) = √3/2. Therefore, cos(7π/6) = -√3/2.

Step 2: Evaluate the Inverse Cosine Function

Now we have cos⁻¹(-√3/2). We need to find an angle θ in the range [0, π] such that cos(θ) = -√3/2. Since cosine is negative, the angle must be in the second quadrant (where x-coordinates are negative).

We know that the reference angle with a cosine of √3/2 is π/6. So, we are looking for an angle in the second quadrant with a reference angle of π/6. To find this angle, we subtract π/6 from π:

θ = π - π/6 = 5π/6

Therefore, cos⁻¹(-√3/2) = 5π/6.

Step 3: Final Answer

Putting it all together:

cos⁻¹(cos(7π/6)) = cos⁻¹(-√3/2) = 5π/6

So, the answer is 5π/6. Easy peasy, right?

Common Mistakes to Avoid

When solving problems like these, it's easy to make a few common mistakes. Here are some things to watch out for:

  1. Forgetting the Range of Inverse Cosine: Always remember that the range of cos⁻¹(x) is [0, π]. If your answer is outside this range, you need to adjust it using reference angles and quadrant rules.
  2. Incorrectly Identifying the Quadrant: Make sure you correctly identify the quadrant in which the angle lies. This will help you determine the correct sign of the trigonometric function.
  3. Confusing Reference Angles: Double-check that you are using the correct reference angle. A simple mistake here can lead to the wrong answer.
  4. Not Simplifying: Always simplify your answer as much as possible. For example, if you end up with an expression like 2π/4, simplify it to π/2.

Practice Problems

To really nail this concept, here are a few practice problems for you to try:

  1. sin⁻¹(sin(4π/3))
  2. tan⁻¹(tan(5π/4))
  3. cos⁻¹(cos(2π/3))
  4. sin⁻¹(sin(7π/6))

Work through these problems step by step, and you'll be solving these types of questions in no time! Remember, practice makes perfect! Grab a piece of paper, work through each problem, and compare your answers with solutions available online or from your textbook.

Tips and Tricks

Here are some extra tips and tricks that can help you solve these problems more efficiently:

  • Use the Unit Circle: The unit circle is your best friend! It provides a visual representation of trigonometric functions and their values at various angles.
  • Memorize Key Values: Memorize the values of sine, cosine, and tangent for common angles like 0, π/6, π/4, π/3, and π/2. This will save you time on exams and homework.
  • Understand Quadrant Rules: Know the signs of sine, cosine, and tangent in each quadrant. This will help you determine the correct sign for your answer.
  • Draw Diagrams: When in doubt, draw a diagram! Sketch the angle on the unit circle, and label the reference angle and coordinates. This can help you visualize the problem and avoid mistakes.
  • Check Your Answer: After you've solved the problem, check your answer to make sure it makes sense. Does it fall within the correct range? Is the sign correct? If something seems off, go back and review your work.

Conclusion

So, there you have it! Solving cos⁻¹(cos(7π/6)) isn't as scary as it looks. By understanding the basics of the unit circle, reference angles, and the range of inverse trigonometric functions, you can solve these problems with confidence. Keep practicing, and you'll become a trigonometric wizard in no time! Remember to avoid common mistakes, use the tips and tricks we discussed, and always double-check your work.

Now that you've mastered this problem, challenge yourself with more complex trigonometric expressions. Explore different angles, different functions, and different combinations. The more you practice, the better you'll become. And don't forget to share your knowledge with your friends and classmates. Together, we can conquer the world of trigonometry! Happy solving, guys!