Problem 23 - Precalculus Essentials

Question

Use a calculator to find the value of each expression rounded to two decimal places. $$\cos ^{-1} \frac{3}{8}$$

Step-by-Step Solution

Answer

The approximate value of \(\cos^{-1}\left(\frac{3}{8}\right)\) rounded to two decimal places using a calculator is found. Ensure that the calculator is in degree mode.

Step 1: Identify the inverse cosine

First, the exercise is asking to find the inverse cosine, or arccos, of \(\frac{3}{8}\). This might also be written as: \(\cos^{-1}\left(\frac{3}{8}\right)\).

Step 2: Compute the inverse cosine function

Next, use a scientific calculator to compute the inverse cosine function. Make sure to check that the calculator is set to degrees mode, not radians. Input the fraction \(\frac{3}{8}\) and press the appropriate button or key to find the inverse cosine.

Step 3: Round to two decimal places

After obtaining the value of the inverse cosine function, round the value to the nearest hundredths place, or two decimal places. This is consistent with the instructions for the exercise.

Key Concepts

Understanding the Arccos Function

The arccosine function, often written as \( \cos^{-1}(x) \), is used to determine the angle whose cosine is \( x \). This is the inverse of the cosine function, which typically helps us find an angle if we know the cosine value. In practical terms, when you see \( \cos^{-1}(\frac{3}{8}) \), it means you are looking for the angle whose cosine is \( \frac{3}{8} \).

• The range of the arccos function is usually from 0 to \( \pi \) radians or 0 to 180 degrees, since cosine is symmetric above the x-axis within these values.
• The domain of arccos is from -1 to 1, since cosine can only yield values in that range.

It's essential to ensure that the input value falls within the permitted domain, otherwise the arccosine might be undefined.

Using a Scientific Calculator

A scientific calculator is handy for solving problems involving trigonometric functions such as arccos. Here’s a brief guide on how you can use a calculator to evaluate \( \cos^{-1}(\frac{3}{8}) \):

• Mode Setting: Ensure the calculator is in the correct mode (degrees or radians) as per the requirements of your task. Degrees are often used if no specific mode is mentioned, as most high school trigonometry uses degree measurement.
• Input: Enter the fraction \( \frac{3}{8} \) directly or as a decimal equivalent (0.375) if your calculator requires it.
• Calculate: Press the "cos^-1" or an equivalent key which stands for the inverse of cosine. Modern calculators often have this function accessible directly or through a shift feature.

After inputting the values, the calculator will give you the angle. The last step is to ensure the result is rounded properly, as per exercise instructions.

Basics of Trigonometry in Finding Angles

Trigonometry is the branch of mathematics that deals with angles, triangles, and their relationships. The concept of inverse trigonometric functions like arccos deals with deducing angles based on known trigonometric values.

• Core Functions: The primary trigonometric functions include sine, cosine, and tangent, traditionally associated with angles in right triangles.
• Inverse Functions: These functions, including arccos, arcsin, and arctan, allow computation of angles when the trigonometric ratios are known.
• Application: Inverse functions like arccos are vital when solving real-world problems involving angles, such as navigation, engineering, and physics.

Finding the arccos of a value is simply one example of applying trigonometry to work backwards from a known ratio (cosine) to discover an unknown angle, thereby solving a range of geometric or practical tasks.