Problem 17 - Algebra 2 and Trigonometry

Question

In \(15-26,\) find each exact value in radians, expressing each answer in terms of \(\pi\) \(y=\arctan 1\)

Step-by-Step Solution

Answer

\( y = \frac{\pi}{4} \)

Step 1: Understanding the Problem

We need to find the angle \( y \) such that \( \arctan(1) = y \). The \( \arctan \) function gives the angle (in radians) whose tangent is the given number (in this case, 1).

Step 2: Recall Tangent Values

Recall that for an angle \( \theta \), \( \tan(\theta) = 1 \) when \( \theta = \frac{\pi}{4} \) since \( \tan \left( \frac{\pi}{4} \right) = 1 \).

Step 3: Express Answer in Radians

The angle \( \theta = \frac{\pi}{4} \) is the angle whose tangent is 1. Therefore, the exact value of \( \arctan(1) \) is \( \frac{\pi}{4} \).

Key Concepts

Radians

Radians are a way to measure angles. They are an alternative to degrees, which is a more familiar unit for many people. In the radian system, the angle of a full circle is equal to \(2\pi\) radians, which corresponds to 360 degrees. This means that to find the value of an angle in radians, you often express it in terms of \(\pi\).

One radian is defined as the angle created when the radius of a circle is wrapped around its edge. This geometrical relationship makes radians particularly useful in many areas of mathematics and physics. In many situations, especially trigonometry and calculus, radians provide cleaner, simpler calculations compared to degrees.

To convert an angle from degrees to radians, you can use the conversion formula:

• \(\text{{Radians}} = \text{{Degrees}} \times \frac{\pi}{180}\)

Radian measures are essential when finding the exact values of trigonometric functions, and often you'll see these values expressed in terms of fractions of \(\pi\).

Tangent

The tangent function, often abbreviated as tan, is a fundamental trigonometric function. In a right triangle, it's defined as the ratio of the opposite side to the adjacent side. Mathematically, it's expressed as:

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

However, beyond triangles, tangent is also represented through the unit circle. The tangent of an angle \(\theta\) in terms of the unit circle is the length of the line segment from the point where the terminal side of the angle meets the circle, projected out to the tangent line.

Tangent values can range from \(-\infty\) to \(+\infty\), unlike sine and cosine, which are limited in range from -1 to 1. It's periodic with a period of \(\pi\) radians, meaning the pattern of the function repeats every \(\pi\) units. This periodicity is crucial in understanding functions like \(\arctan\), the inverse of tangent, which limits its output or range of angles to \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) radians.

Exact Value

When working with trigonometric functions, finding the exact value of an expression means expressing the solutions in precise mathematical terms rather than as an approximate decimal. The exact values can typically be represented with fractions, square roots, or terms involving \(\pi\).

In the context of trigonometric functions like \(\tan\), finding the exact value means identifying the angle that produces the given tangent ratio. For example, when tasked with finding \(\arctan(1)\), you are looking to determine the angle for which the tangent is exactly 1. We know that this occurs at \(\theta = \frac{\pi}{4}\) because \(\tan\left(\frac{\pi}{4}\right)=1\).

Using exact values is particularly important in theoretical mathematics, as it enables precise calculations in further math problems or in the accuracy of expressions derived in calculus, algebra, and trigonometry. When angles are expressed exactly, especially in terms of \(\pi\), they help maintain this level of precision throughout mathematical evaluations.