Problem 13 - Algebra 2 and Trigonometry

Question

In \(3-14,\) find each value of \(\theta : \mathbf{a} .\) in degrees \(\mathbf{b} .\) in radians $$ \theta=\arctan 0 $$

Step-by-Step Solution

Answer

\( \theta = 0^\circ \) or \( 0 \) radians.

Step 1: Understanding the Arctangent Function

The arctangent function, denoted as \( \arctan \), is the inverse of the tangent function. It gives the angle \( \theta \) for which the tangent of \( \theta \) is a particular value. Here, we need to find \( \theta \) such that \( \tan(\theta) = 0 \).

Step 2: Determine \( \theta \) in Degrees

Since \( \tan(0^\circ) = 0 \), the angle in degrees for which tangent of \( \theta \) is 0 is \( \theta = 0^\circ \). This is because the tangent of 0 degrees is indeed zero.

Step 3: Convert \( \theta \) into Radians

Knowing that \( \theta = 0^\circ \) corresponds to 0 radians (because \( 0 \times \frac{\pi}{180} = 0 \)), the angle \( \theta \) in radians is the same, \( 0 \) radians.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find angles when we know the value of the trigonometric function. Specifically, the arctangent function, denoted by \( \arctan \), helps us find an angle \( \theta \) such that \( \tan(\theta) \) equals a known value. This is particularly useful because in many trigonometric problems, we know the outcome of a trigonometric ratio but need to determine the corresponding angle.

For example, solving \( \theta = \arctan(0) \) requires us to find the angle where the tangent is zero. In simpler terms, it is asking "what angle has a tangent of 0?" Since the tangent function is the ratio of the opposite to the adjacent side in a right triangle, the value is zero when the opposite side is 0, which occurs at \( 0^\circ \). Understanding how inverse functions work is crucial because they allow us to go from a trigonometric value back to an angle, aiding in solving trigonometric equations.

Angle Conversion

Converting between degrees and radians is a fundamental skill in trigonometry. Angles can be expressed in different units: degrees or radians. One full rotation around a circle is \( 360^\circ \) in degrees and \( 2\pi \) in radians. To convert degrees to radians, we use the conversion factor \( \frac{\pi}{180} \).

Let's apply this to our exercise: we found \( \theta = 0^\circ \). To convert this angle to radians, we multiply by the conversion factor
\[ \theta_{radians} = 0^\circ \times \frac{\pi}{180} = 0 \text{ radians} \]

As with many things in math, understanding the concept behind conversions is as important as memorizing the formula. It allows flexibility and accuracy when solving broader mathematical problems in different units.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables involved. These identities are tools that simplify complex expressions and equations, making calculations more manageable. In our context, the identity \( \tan(0^\circ) = 0 \) helps us determine \( \theta \) when finding \( \arctan(0) \).

One powerful aspect of trigonometric identities is their ability to help verify solutions. By using known identities such as \( \tan(-x) = -\tan(x) \) and the fact that \( \tan(180^\circ + x) = \tan(x) \), we can ensure calculations are correct and also extend them to other problems like \( \tan(\pi) = 0 \).

Learning and applying these identities help by creating shortcuts in calculations, allowing us to solve problems much quicker and with a deeper understanding of the relationships between angles and their trigonometric values.