Problem 25 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a calculator to approximate the value of the expression, if possible. Round your answer to the nearest hundredth. arccos( -0.9 )

Step-by-Step Solution

Answer

The arccos(-0.9) rounded off to the nearest hundredth is approximately 154.50 degrees.

Step 1: Calculate arccos(-0.9)

Use a scientific calculator and find the inverse cosine function, arccos, of -0.9. Most scientific calculators will have the arccos function.

Step 2: Round the result

Round off the result obtained from Step 1 to the nearest hundredth.

Key Concepts

arccos function

The arccos function, known as the inverse cosine function, is a type of inverse trigonometric function. It allows us to find the angle whose cosine value is a given number. Essentially, when you're given \(x\) and asked for what angle \(\theta\) this corresponds to such that \( ext{cos}(\theta) = x\), you use the arccos function. It's important to understand that the arccos function only deals with values ranging from \(-1\) to \(1\), hence it will only accept numbers within this range. For example, if you're finding \( ext{arccos}(-0.9)\), it yields the angle, if you imagine it, in radians or degrees, where cosine would give \(-0.9\). This function typically provides results in the interval from 0 to \(\pi\) radians, or from 0° to 180°. Understanding this range is crucial because it helps ensure the correct angle results fit within the conventional circle of solutions.

scientific calculator usage

Using a scientific calculator may seem intimidating, but it's actually straightforward once you know which buttons to press. Most modern scientific calculators have a dedicated button for trigonometric functions. You will typically find these labeled as `sin`, `cos`, and `tan`. For our purposes, you need the inverse functions, often accessed by pressing a `shift` or `inv` button in conjunction with the `cos` button to activate the `arccos` function.

Here’s how you can perform the calculation of \( ext{arccos}(-0.9)\):

• Locate and press the `cos⁻¹` or `arccos` button. Often, you need to press `Shift` before `cos` to access this function.
• Input `-0.9` and then press `=` or `Enter`. This will give you the angle, \( heta\), for which \( ext{cos}(\theta) = -0.9\).

Now, you have used your scientific calculator to determine the angle using the arccos function. The key is understanding the layout and functionality of the calculator. Practicing with your calculator on various exercises helps build comfort and familiarity.

rounding numbers

Rounding numbers is a basic, yet essential arithmetic skill. It involves adjusting the value of a number to make it simpler or to fit certain specified criteria, such as decimal places. In this context, we're focusing on rounding to the nearest hundredth place. To successfully round a number to the nearest hundredth, you follow these steps:

First, identify the decimal figure at the hundredth place. This is the second number to the right of the decimal. For instance, in 1.236, the digit `3` is in the hundredth place.
Next, check the digit immediately to the right — the thousandth place. If this digit is 5 or higher, increase the hundredth place digit by one. If it is 4 or lower, leave the hundredth place digit as is.

• For example, 1.236 would round to 1.24 because the `6` in the thousandth place prompts an increase in the `3` in the hundredth place.
• Conversely, for a value like 1.234, it remains 1.23 because the `4` doesn't warrant a change.

Accurately rounding numbers ensures precision in calculations and is particularly important when dealing with measurements or detailed data presentations.