Problem 10 - Algebra 2

Question

Find each angle measure to the nearest tenth of a degree. \(\tan ^{-1} 0.3333\)

Step-by-Step Solution

Answer

The nearest tenth of a degree for \( \tan^{-1} 0.3333 \) is approximately \( 18.4^{\circ} \).

Step 1: Using the Inverse Tangent Function

The exercise calls for calculating the value of \( \tan^{-1} (0.3333) \). Since \( \tan^{-1} \) is the inverse function of tangent, what we are actually trying to find is the angle whose tangent is 0.3333.

Step 2: Ensure Calculator is in Degree Mode

Ensure that the calculator is in degree mode as the question asks for the answer in degrees. Most scientific and graphing calculators will have options to switch between degree and radian mode.

Step 3: Calculate the Angle Measure

Using the calculator in degree mode, calculate \( \tan^{-1} (0.3333) \). This will give the angle measure.

Step 4: Round to the Nearest Tenth

Once you get the result from your calculator, round it to the nearest tenth as per the question's instructions.

Key Concepts

Tangent Function

The tangent function is one of the fundamental concepts in trigonometry. It's part of the set of functions known as trigonometric functions, which include sine and cosine. The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. This can be expressed as:

• \[\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}\]

Trigonometric functions are vital in mathematics and engineering because they relate to angles and dimensions in geometrical figures and functions.
The inverse tangent function, written as \( \tan^{-1} \) or \( \arctan \), reverses the process: given a tangent value, it returns the original angle. For example, finding \( \tan^{-1}(0.3333) \) means identifying the angle whose tangent is 0.3333. It's important in all fields where angle calculation and transformations are required, like computer graphics, navigation, and signal processing.

Degree Mode

Almost all calculators can operate in two primary modes when dealing with angles: degree mode and radian mode. Degree mode is an alternative that many people are comfortable with because it's based on the familiar 360 degrees in a circle.
When solving problems involving trigonometric functions like the tangent, it's critical to ensure your calculator is set to degree mode if the question specifically requests the answer in degrees. This mode will provide results that align with the way angles are often taught in high schools and used in everyday life.
Switching between modes is usually straightforward in most calculators:

• Find the mode setting function.
• Select the option to change between radian or degree mode.
• Ensure the display indicates the correct mode before calculating.

Using degree mode ensures your results make sense when compared with standard angles like 30°, 45°, or 60°.

Radian Mode

Radian mode is the other common way of expressing and calculating angles, and it is often used in more advanced mathematics, particularly calculus. Instead of using 360 degrees to mark a full circle, radians use the concept where the length of the arc of a circle's radius defines the angle.
One complete circle is equivalent to \( 2\pi \) radians, or about 6.28318 radians. It's a natural fit for calculations involving equations of circles and other curves.
While radian mode might seem less intuitive at first, it provides a seamless approach to tackle problems involving rotations, wave functions, and periodic behaviors. When you're working with a calculator:

• Ensure you understand whether you should use radian mode based on the problem's context.
• Many scientific applications, including physics and engineering, prefer radians for precision.
• Converting between degrees and radians can be completed using the relation: \( 1^{\circ} = \frac{\pi}{180} \) radians.

Both radian and degree modes have their places in problem-solving, depending on the specific needs and conventions of the context.