Problem 26 - Algebra and Trigonometry

Question

Use a calculator to find the value of each expression rounded to two decimal places. $$ \cos ^{-1} \frac{\sqrt{7}}{10} $$

Step-by-Step Solution

Answer

After calculation and rounding, the solution will be given based on your calculator's computation. Note that the actual answer varies depending on the settings of your calculator (degrees or radians).

Step 1: Expression Formation

Take the given expression \( \cos^{-1} \frac{\sqrt{7}}{10} \). We need to compute the arccosine, or the inverse cosine, of \( \frac{\sqrt{7}}{10} \). In other words, we are finding the angle whose cosine is \( \frac{\sqrt{7}}{10} \).

Step 2: Calculation

On your calculator, first calculate \( \frac{\sqrt{7}}{10} \). Then, find the inverse cosine of the result. Make sure the calculator is set to the correct mode (degrees or radians) depending on the requirements.

Step 3: Rounding Off

Round off the result from your calculator to two decimal places. This is the final answer.

Key Concepts

Understanding Arccosine

The term **arccosine**, often written as \( \cos^{-1}(x) \), refers to the inverse function of the cosine. It is used to determine the angle when the cosine value is known. Simply put, given a cosine value, arccosine will help us find the corresponding angle. This function is very important in trigonometry because it provides a way to go from trigonometric ratios back to angles.

Here’s a simple way to think about it:

• Imagine you have a ladder leaning against a wall. You know how far the top of the ladder is from the wall (the adjacent side), and you know how long the ladder is (the hypotenuse). The arccosine helps you find the angle between the ladder and the ground.
• The result of the arccosine function is typically given in radians or degrees.

When you see \( \cos^{-1} \), remember you are looking for an angle, which is why it is crucial to set your calculator correctly, as mentioned in the exercise.

Rounding Numbers

Rounding numbers is a mathematical step used to simplify numbers or make them easier to understand by limiting the number of decimal places. In many math problems, especially those involving calculations, rounding is essential to obtain a handleable figure and also when precision beyond a certain point is not necessary.

Here is a quick guideline on how to round numbers to two decimal places:

• Check the third decimal digit.
• If it is 5 or greater, you round the second digit up by one.
• If it is less than 5, you leave the second digit as it is.

In this particular exercise, after calculating the arccosine value using a calculator, the result is rounded to two decimal places to provide a concise and useful answer. Rounding helps in keeping answers neat and in line with general math expectations, especially in educational settings.

Calculator Usage

Calculators are handy tools to solve mathematical problems quickly and efficiently. In problems involving inverse trigonometric functions like arccosine, calculators become almost indispensable due to their complexity and the need for precision.

Here are some steps to use your calculator for finding an arccosine:

• Ensure your calculator is in the correct mode. Decide if your answer should be in radians or degrees, and switch the settings accordingly.
• Input the expression properly, starting with the division if needed. In the exercise case, you first calculate \( \frac{\sqrt{7}}{10} \).
• Next, locate and use the inverse cosine function, usually marked as \( \cos^{-1} \) or "ACOS" on calculators.

Finally, after receiving the result, use your calculator's rounding function if available, or round the result manually to ensure the answer meets the problem requirements. Mastery of calculators, whether basic or scientific, can greatly simplify and speed up your work.