Problem 22 - Algebra 2 and Trigonometry

Question

In \(15-26,\) find each exact value in radians, expressing each answer in terms of \(\pi\) \(y=\arctan \sqrt{3}\)

Step-by-Step Solution

Answer

The exact value of \( y \) is \( \frac{\pi}{3} \).

Step 1: Understand the Problem

We are asked to find \( y = \arctan \sqrt{3} \). This means we need to find the angle \( y \) whose tangent is \( \sqrt{3} \).

Step 2: Recall Trigonometric Ratios

Remember that the tangent of \( \frac{\pi}{3} \) radians is \( \sqrt{3} \). So, \( \tan \frac{\pi}{3} = \sqrt{3} \).

Step 3: Solve for the Angle

Since \( \tan y = \sqrt{3} \), and \( \tan \frac{\pi}{3} = \sqrt{3} \), the value of \( y \) must be \( \frac{\pi}{3} \). Thus, \( y = \frac{\pi}{3} \).

Step 4: Verify the Interval

The range of \( \arctan \) is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), which includes \( \frac{\pi}{3} \), so the solution \( y = \frac{\pi}{3} \) is correct.

Key Concepts

Understanding the Arctan Function

The arctan function, also known as the inverse tangent function, is a critical concept in trigonometry. Its primary role is to determine the angle whose tangent is a given number. This function is especially crucial because it helps us work backwards from a trigonometric ratio to find the corresponding angle.
When you see the problem asking for \( y = \arctan \sqrt{3} \), it's asking us to find an angle \( y \) where the tangent of \( y \) equals \( \sqrt{3} \). The arctan function has a specific range or interval of values it can output, which is between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) radians. This interval ensures that every output of the function corresponds to a unique input value.

• The domain of the arctan function is all real numbers, meaning you can input any real number and the function will give back an angle.
• The output or range is tailored to fit within \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), making the function reversible and precise.

Delving into Trigonometric Ratios

Trigonometric ratios are fundamental in linking angles to side lengths in right triangles. The primary trigonometric ratios are sine, cosine, and tangent. Here, our focus is on the tangent ratio, which gives us the relationship between the opposite side and the adjacent side of an angle in a right triangle.
For an angle \( \theta \), the trigonometric ratio can be expressed as:

• \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)

Knowing these ratios is vital as they allow us to decode the inverse functions easily, such as \(\arctan\). Often, through memorization or quick reference, we might know that the tangent of certain angles results in specific values:

• From trigonometric tables, \( \tan \frac{\pi}{3} = \sqrt{3} \).
• This information confirms that when asked for \( \arctan \sqrt{3} \), the angle in question is \( \frac{\pi}{3} \).

Exploring the Tangent Function

The tangent function plays a pivotal role in trigonometry. It interprets the ratio of two sides in a triangle to an angle measurement. Specifically, in a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side.
When calculating tangent values, think of the following examples:

• In a triangle where the opposite side is twice the length of the adjacent side, \( \tan(\theta) = 2 \).
• This function can sometimes produce values that are not easy to intuitively grasp, such as \( \tan(\frac{\pi}{3}) = \sqrt{3} \).

The tangent function has a periodic nature and repeats every \(\pi\) radians. Its period helps determine angles beyond the standard \(0\) to \(\pi/2\) range. Understanding this cyclic behavior is valuable for extending the tangent concept across different quadrants and exploring more advanced trigonometric applications.