Question

Use a calculator to approximate each value. $$ \arccos (0.6341) $$

Short answer

Using a calculator, \( \arccos(0.6341) \approx 50.57^\circ \) or \( 0.883 \) radians.

Step-by-step solution

Understand the Function

The function \( \arccos \) refers to the inverse cosine function. It gives the angle whose cosine is the given number. In this problem, we need to find the angle \( x \) such that \( \cos(x) = 0.6341 \).

Set Up the Problem

We need to approximate \( \arccos(0.6341) \) using a calculator. This means we will use the calculator's inverse cosine function, often denoted as \( \cos^{-1} \), to find the angle in radians or degrees.

Calculate the Value

Using your calculator, enter \( 0.6341 \) and then apply the \( \cos^{-1} \) or "arccos" function. Ensure your calculator is set to the correct unit (radians or degrees) as desired for the final answer.

Interpret the Result

The calculator will output a value. Ensure you are interpreting this value in the correct unit based on your calculator's settings (either radians or degrees). This value is the angle whose cosine is approximately \( 0.6341 \).

Key concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are a group of functions that reverse the traditional trigonometric functions such as sine, cosine, and tangent. They are used to find the angle when the value of the trigonometric function is known. Since regular trigonometric functions are periodic and not one-to-one, their inverses are defined over restricted domains:

• The inverse sine function, denoted as \( ext{arcsin} \, or \, \sin^{-1} \), is only defined for input values between -1 and 1, providing output values from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \) radians.
• The inverse cosine function, \( ext{arccos} \, or \, \cos^{-1} \), is also defined for input values between -1 and 1, with output values ranging from 0 to \( \pi \) radians.
• The inverse tangent function, \( ext{arctan} \, or \, \tan^{-1} \), can handle any real number input and yields output values between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \) radians.

Understanding these domains and ranges is crucial when solving problems involving inverse trigonometric functions.

Arccos

The term "arccos" refers to the inverse of the cosine function. It helps find the angle whose cosine is a specific value. For instance, if you have a cosine value like 0.6341, the arccos function will provide you the angle \( x \) such that \( \cos(x) = 0.6341 \).
When you find \( ext{arccos}(0.6341) \), you are looking for an angle within the range from 0 to \( \pi \) radians. This function is essential in trigonometry because it bridges the gap between angle measurements and trigonometric values.
Its applications are vast and include fields such as engineering, physics, and even computer graphics.

• Provides a method to solve for angles in various geometric problems.
• Useful in analyzing oscillatory systems such as waves.
• Vital for converting between coordinate systems.

Calculator Usage

A calculator is an indispensable tool for finding the value of inverse trigonometric functions like arccos. When tasked with computing \( ext{arccos}(0.6341) \), a calculator simplifies the process greatly:
To start:

• Ensure your calculator is on and is in the correct mode—either degrees or radians. This setting affects the output and should match the requirements of your problem.
• Enter "\( 0.6341 \)" into the calculator.
• Use the inverse cosine function, typically denoted as \( ext{cos}^{-1} \) or "arccos" on your calculator. Most scientific calculators have a dedicated button for this function.

After these steps, the calculator will give you the desired angle. Double-check the units of the result (radians or degrees) to ensure you're interpreting the answer correctly.

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It relates the angle in a right triangle to the ratio of the length of the adjacent side over the hypotenuse. In the unit circle, cosine represents the horizontal coordinate of a point moved counterclockwise from the positive x-axis.
Key points about cosine:

• The cosine of an angle \( \theta \) in a right triangle is given by \( \cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} \).
• In the unit circle, \( \cos(\theta) \) is the x-coordinate of the point on the circle at an angle \( \theta \) radians from the positive x-axis.
• Cosine values range from -1 to 1. Positive values indicate that the angle is between \( 0 \) and \( \frac{\pi}{2} \) or between \( \frac{3\pi}{2} \) and \( 2\pi \).

Understanding the behavior of the cosine function is vital when working with its inverse, the arccos function, especially in calculating angles based on known cosine values.