Solving Arccosine Expressions: A Step-by-Step Guide

Hey math enthusiasts! Let's dive into solving the expression: $\arccos (-1) + \arccos \frac{1}{2} + 1.4$. This problem involves understanding the arccosine function and applying it to find the final value. We'll break it down step-by-step, making sure everything is clear and easy to follow. Get ready to flex those math muscles!

Understanding the Arccosine Function and Initial Steps

First off, what exactly is arccosine? Well, guys, arccosine (often written as $\arccos$ or $\cos^{-1}$) is the inverse function of the cosine function. It answers the question: "What angle has a cosine equal to this value?" The arccosine function gives you an angle, and the answer is always in radians or degrees. The output range for $\arccos$ is usually between 0 and $\pi$ radians (or 0 and 180 degrees).

Let’s start with the given expression: $\arccos (-1) + \arccos \frac{1}{2} + 1.4$. Our mission is to find the value of this. We will start by finding the values of the individual arccosine terms.

Calculating $\arccos(-1)$

So, we need to find the angle whose cosine is -1. Think about the unit circle, folks. Cosine corresponds to the x-coordinate. Where on the unit circle is the x-coordinate equal to -1? That's at an angle of $\pi$ radians (or 180 degrees). Therefore, $\arccos(-1) = \pi$. As a decimal, this is approximately 3.14. We will use two decimal places for intermediate calculations, so we will use 3.14.

Calculating $\arccos \frac{1}{2}$

Next up, we need to find the angle whose cosine is $\frac{1}{2}$. On the unit circle, the x-coordinate is $\frac{1}{2}$ at an angle of $\frac{\pi}{3}$ radians (or 60 degrees). So, $\arccos \frac{1}{2} = \frac{\pi}{3}$. Converting to decimal, $\frac{\pi}{3}$ is approximately 1.05 (rounded to two decimal places).

Combining the Results and Final Calculation

Now that we have the individual values of the arccosine terms, we can substitute these back into the original expression and find the final answer. We have $\arccos (-1) = 3.14$ and $\arccos \frac{1}{2} = 1.05$. Let’s put it all together.

Putting it all together

So, our expression becomes $3.14 + 1.05 + 1.4$. Let's add these numbers together. First, add 3.14 and 1.05, which gives you 4.19. Now, add 1.4 to 4.19. This results in 5.59.

Rounding to the Nearest Tenth

Finally, we're asked to round our answer to the nearest tenth. The number 5.59 rounded to the nearest tenth is 5.6. So, the final answer to the expression $\arccos (-1) + \arccos \frac{1}{2} + 1.4$, rounded to the nearest tenth, is 5.6.

Detailed Step-by-Step Breakdown

Here’s a summary of the steps to make sure everything is crystal clear:

1. Understand Arccosine: Remember that arccosine gives you the angle whose cosine is a given value.
2. Calculate $\arccos(-1)$: Find the angle where the cosine is -1. This is $\pi$ radians, or approximately 3.14.
3. Calculate $\arccos(\frac{1}{2})$: Find the angle where the cosine is $\frac{1}{2}$. This is $\frac{\pi}{3}$ radians, or approximately 1.05.
4. Substitute and Add: Replace the arccosine terms in the original expression with their values: $3.14 + 1.05 + 1.4 = 5.59$
5. Round to the Nearest Tenth: Round 5.59 to the nearest tenth: 5.6.

So, there you have it! We've successfully navigated the expression step-by-step. Keep practicing, and you'll get the hang of these problems in no time. If any of this felt confusing, re-read the parts that weren't clear to you. Math is all about practice and understanding the basics.

Tips for Success and Further Exploration

To really nail these kinds of problems, here are some tips:

• Memorize Key Values: Knowing the cosine and sine values for common angles (0, $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, $\frac{\pi}{2}$, and their multiples) is super helpful. This allows you to quickly evaluate many trigonometric functions without constantly referring to the unit circle or a calculator.
• Use the Unit Circle: The unit circle is your best friend when it comes to understanding trigonometric functions. It visually represents the relationships between angles and their cosine and sine values. Spend some time studying it, and you'll find it incredibly useful.
• Practice, Practice, Practice: The more problems you solve, the more comfortable you’ll become. Work through different examples to solidify your understanding.
• Explore Inverse Trig Functions: Besides arccosine, there are also arcsine and arctangent. Try solving problems involving these functions too, to broaden your understanding of inverse trigonometric functions.
• *Use a Calculator: A scientific calculator can be a great tool, especially for checking your work and for calculations. Make sure you know how to switch between degrees and radians on your calculator. You want to make sure the unit on the calculator matches the question.

I hope you enjoyed this guide. Keep up the great work, and don't be afraid to ask questions. Happy calculating, everyone!