Problem 55 - Algebra 2

Question

Find each angle measure to the nearest tenth of a degree. $$ \tan ^{-1} \sqrt{2} $$

Step-by-Step Solution

Answer

The angle measure is approximately 54.7 degrees.

Step 1: Recalling the Definition of Inverse Tangent

The inverse tangent or arctangent (written as \( \tan^{-1} \) or atan) of a value tells us the angle whose tangent is that value. Here we are asked to find the angle who’s tangent is \( \sqrt{2} \).

Step 2: Using Calculator to Find Angle Measure

Enter the value \( \sqrt{2} \) in the calculator and press the atan or \( \tan^{-1} \) button. This gives us the result in radians. To convert it into degrees, multiply the result by \( \frac{180}{\pi} \).

Step 3: Rounding the Result

Lastly, ensure that the resulting value is rounded to the nearest tenth as per the exercise instructions.

Key Concepts

Tangent

The tangent function is one of the primary trigonometric functions, and it relates the angles of a right triangle to the ratios of two of its sides. Specifically, for an angle in a right triangle:

• The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
• In mathematical terms, if we have an angle \( \theta \), then \( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \).

This function, along with sine and cosine, forms the foundation of trigonometry. It's important to understand this relationship so we can use the inverse tangent, or \( \tan^{-1} \), to find an angle given the length ratio.
When you operate on a value using \( \tan^{-1} \), you are finding the angle whose tangent is the given value.
The inverse tangent function is crucial for problems where the angle isn't directly available but the side ratios are known, such as when calculating the arctangent of \( \sqrt{2} \) in our problem.

Angle Measurement

Angle measurement is fundamental in trigonometry, which is the branch of mathematics dealing with triangles, particularly right-angled triangles. Angles are most commonly measured in degrees or radians:

• Degrees split a full rotation into 360 equal parts.
• Radians measure the angle by the length of the arc that it cuts on a circle's circumference, with the full circle containing \( 2\pi \) radians.

Understanding how to switch between these measurements is key because different contexts or tools (like calculators or engineering tools) might use different units.
In this particular problem, you need to find an angle using the inverse tangent which originally gives an output in radians. This makes it necessary to convert the result into degrees for common interpretability.

Degree Conversion

Degree conversion is the process of changing an angle measurement from radians to degrees. This is essential in ensuring that angle values align with most practical applications, as degrees are often more intuitive to understand.

• The conversion relies on the relationship \( 180^\circ = \pi \) radians.
• To convert radians to degrees, you multiply by \( \frac{180}{\pi} \).

Using this conversion factor, if you start with an angle in radians, multiplying by \( \frac{180}{\pi} \) will yield the angle in degrees.
This conversion was used in our exercise to find the angle measure after determining the inverse tangent of \( \sqrt{2} \), ensuring the solution was in the desired units.
Accurate conversion is necessary to meet problem requirements, like getting the final answer in degrees and rounding it appropriately.