Evaluate Tan⁻¹(tan(3π/4)): Step-by-Step Solution

by ADMIN 49 views

Hey guys! Let's dive into a fun math problem today: evaluating the expression tan⁻¹(tan(3π/4)). This might seem tricky at first, but don't worry, we'll break it down step by step. We'll explore the concepts of inverse trigonometric functions and the unit circle to solve this problem. So, buckle up, and let’s get started!

Understanding the Problem

Before we jump into the solution, let’s make sure we understand what the question is asking. We are given the expression tan⁻¹(tan(3π/4)) and we need to find its exact value. The key here is to understand the relationship between the tangent function and its inverse, the arctangent function (tan⁻¹). Remember, the arctangent function gives us the angle whose tangent is a given value. However, we need to be mindful of the range of the arctangent function, which plays a crucial role in finding the correct answer. Let’s dive into each step to make it clearer.

Step 1: Evaluate the Inner Tangent Function

First, we need to find the value of the inner part of the expression, which is tan(3π/4). To do this, we can use our knowledge of the unit circle. The angle 3π/4 is in the second quadrant, where the tangent function is negative. Specifically, 3π/4 is a reference angle of π/4 away from the π axis. We know that tan(π/4) = 1, so tan(3π/4) = -1. Therefore, we’ve simplified our expression to tan⁻¹(-1).

Understanding the unit circle is super important here. The unit circle helps visualize the trigonometric functions for different angles. In the second quadrant, sine is positive, while cosine and tangent are negative. This understanding helps us determine the sign of tan(3π/4).

Step 2: Evaluate the Arctangent Function

Now we need to find the angle whose tangent is -1. This is where the arctangent function, tan⁻¹, comes in. The range of the arctangent function is (-π/2, π/2), which means the output angle must lie between -π/2 and π/2. So, we are looking for an angle within this range that has a tangent of -1.

We know that tan(-π/4) = -1, and -π/4 falls within the range of the arctangent function. Therefore, tan⁻¹(-1) = -π/4. This is a crucial step. The range restriction of the arctangent function ensures that we get a unique solution. If we didn't consider the range, we might end up with an incorrect answer. The arctangent function essentially undoes the tangent function, but within its defined range.

Step 3: Final Answer

So, after evaluating both the inner tangent function and the outer arctangent function, we find that tan⁻¹(tan(3π/4)) = -π/4. This is the exact value of the expression. It's important to note that if we had simply canceled out the tan and tan⁻¹, we might have incorrectly concluded that the answer is 3π/4. However, since 3π/4 is not within the range of the arctangent function, this would be incorrect. Always remember to check the range of the inverse trigonometric functions!

Key Concepts Revisited

Let's quickly recap the key concepts we used to solve this problem:

  1. Unit Circle: We used the unit circle to find the value of tan(3π/4). Understanding the signs and values of trigonometric functions in different quadrants is essential.
  2. Tangent Function: We evaluated tan(3π/4), recalling that tangent is negative in the second quadrant.
  3. Arctangent Function (tan⁻¹): We used the arctangent function to find the angle whose tangent is -1. It's crucial to remember that the range of arctangent is (-π/2, π/2).
  4. Range of Inverse Trigonometric Functions: The range restriction of the arctangent function is what allowed us to find the unique and correct solution.

Why is the Range of Arctangent Important?

You might be wondering, why all the fuss about the range of the arctangent function? Well, the range is critical because it ensures that the arctangent function has a unique output for each input. Think about it this way: the tangent function is periodic, meaning it repeats its values at regular intervals. For example, tan(x) = tan(x + π) = tan(x + 2π), and so on. So, if we didn't restrict the range of the arctangent function, there would be infinitely many possible answers for tan⁻¹(-1). By defining the range as (-π/2, π/2), we ensure that there is only one correct answer, which is -π/4 in our case.

This concept applies to other inverse trigonometric functions as well. For instance, the range of arcsine (sin⁻¹) is [-π/2, π/2], and the range of arccosine (cos⁻¹) is [0, π]. These range restrictions are essential for the inverse trigonometric functions to be well-defined.

Common Mistakes to Avoid

When dealing with inverse trigonometric functions, it's easy to make mistakes if you're not careful. Here are a few common pitfalls to avoid:

  • Ignoring the Range: As we've seen, the range of inverse trigonometric functions is crucial. Don't forget to check if your answer falls within the correct range.
  • Canceling Functions Prematurely: It might be tempting to simply cancel out tan and tan⁻¹ in expressions like tan⁻¹(tan(x)), but this can lead to incorrect answers if x is not within the range of the arctangent function. Always evaluate the inner function first.
  • Not Using the Unit Circle: The unit circle is your best friend when dealing with trigonometric functions. It helps you visualize angles and their corresponding trigonometric values. Make sure you're comfortable using it.

Practice Problems

To solidify your understanding, let's try a few similar problems:

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

Try solving these on your own, and remember to consider the range of the inverse trigonometric functions. Working through these practice problems will help you become more confident in dealing with these types of expressions. Grab a pen and paper, and let's get to it!

Real-World Applications

You might be wondering, where do these trigonometric and inverse trigonometric functions come in handy in the real world? Well, they're used in a wide variety of fields, including:

  • Physics: Trigonometry is essential for describing wave motion, oscillations, and the behavior of light and sound.
  • Engineering: Engineers use trigonometry to design structures, analyze forces, and navigate aircraft and ships.
  • Computer Graphics: Trigonometric functions are used to create realistic images and animations in computer graphics.
  • Navigation: Trigonometry is used in GPS systems and other navigation technologies to determine position and direction.

So, understanding these concepts is not just about solving math problems; it's about gaining valuable skills that can be applied in many different areas.

Conclusion

Alright, guys, we've successfully evaluated the expression tan⁻¹(tan(3π/4)) and found the exact value to be -π/4. We've also discussed the importance of understanding the range of inverse trigonometric functions and how the unit circle can help us solve these types of problems. Remember, the key to mastering these concepts is practice, so keep working on those practice problems and don't hesitate to ask questions if you get stuck. Math can be challenging, but with a little effort and the right approach, you can conquer any problem. Happy calculating, and see you next time!