Problem 23 - Precalculus: A Concise Course

Question

Use a calculator to evaluate the expression. Round your result to two decimal places. $$ \arccos 0.37 $$

Step-by-Step Solution

Answer

The arccosine of 0.37, rounded to two decimal places, is 1.16 radians.

Step 1: Setting the Calculator

First, ensure that the scientific calculator is set in radian mode. This is necessary as the arccos function operates within the radian measurement system.

Step 2: Input the Value

Enter the value 0.37 into the calculator’s arccos function. This can typically be performed by inputting 0.37 then pressing the arccos button.

Step 3: Evaluate the Expression

Calculate the value of the arccos function with 0.37. The calculator will give a result.

Step 4: Rounding the Result

Finally, round the resulted number to two decimal places as directed in the exercise

Key Concepts

Arccosine Function

The arccosine function, often written as \( \arccos \), is an inverse trigonometric function. It is used to determine the angle whose cosine is a given number. In other words, \( \arccos(x) \) will give you an angle \( \theta \) such that \( \cos(\theta) = x \). This is very useful when you know a cosine value and want to figure out the corresponding angle.

The range of the arccosine function is from 0 to \( \pi \) radians, which means it will give you an angle that lies within this interval. This limitation ensures that the arccosine function is properly defined, as cosine is not one-to-one over its entire range. It's important to know that the input value for \( \arccos \) should be between \(-1\) and 1, since these are the possible values for the cosine of an angle.

Radians

Radians are a way of measuring angles based on the radius of a circle. Unlike degrees, which break a circle into 360 equal parts, radians measure angles as the length of an arc divided by the radius of the circle. This makes radians a natural and powerful way to describe angles in mathematics. For instance, a full circle is 360 degrees or \( 2\pi \) radians.

When using trigonometric functions like \( \arccos \), it is common to use radians instead of degrees. Many scientific calculators have the capability to switch between degree and radian modes. It is crucial to ensure the calculator is set to the correct mode, as this will affect your calculations and results.

Scientific Calculator

Scientific calculators are essential tools for solving mathematical problems that involve trigonometric functions like \( \arccos \). They allow users to perform complex calculations with ease. These calculators have different mode settings – typically degrees and radians – so make sure to select the correct one for your task.

To evaluate \( \arccos 0.37 \) using a scientific calculator, follow these steps:

• Set the calculator to "radian" mode.
• Enter 0.37.
• Press the \( \arccos \) function button.

After performing these steps, your calculator will display the result in radians.

Rounding Numbers

Rounding numbers helps in simplifying and presenting them in a more concise way. Rounding to two decimal places is often used in mathematics to make results more readable without losing significant detail.

When rounding, you look at the third decimal place. If it is 5 or greater, increase the second decimal place by one. If it is less than 5, keep the second decimal the same.

For example, suppose the result from the calculator for \( \arccos 0.37 \) is 1.1892 radians. To round it to two decimal places, examine the third decimal place, which is 8. Since it's greater than 5, the second decimal increases from 8 to 9, thus giving a rounded result of 1.19 radians. This step is crucial, especially in mathematical computations where precision is necessary.