Problem 10 - Precalculus with Limits

Question

Evaluate the expression without using a calculator. \(\arctan 1\)

Step-by-Step Solution

Answer

The value of \( \arctan 1 \) is \( 45° \)

Step 1: Identify the given value's counterpart standard angle

We are given 1 as the argument for the arctan function. We have to find a standard angle which has its tangent equal to 1. The standard angles in trigonometry usually are: 0°, 30°, 45°, 60° and 90°.

Step 2: Evaluate the tangent of the standard angles

Evaluate the tangent of the standard angles until we find a value that equals 1. \(\tan 0° = 0\), \(\tan 30° ≠ 1\), \(\tan 45° = 1\). So, 45° is our needed angle.

Step 3: Apply the arctan function rule

Since we have found that 45° is the angle whose tangent equals to 1, we can get the value of \(\arctan 1 = 45°\) according to the inverse function rule.

Key Concepts

Tangent Function

The tangent function is one of the basic trigonometric functions and is defined as the ratio of the opposite side to the adjacent side in a right triangle. You can think of it as a comparison between these two sides. In the context of a unit circle, where the circle's radius is 1, the tangent of an angle is the y-coordinate divided by the x-coordinate. This function is periodic with a period of 180° or \(\pi\) radians.

When you evaluate the tangent for standard angles like 0°, 30°, 45°, 60°, and 90°, each angle has its unique value. For example:

• \(\tan 0° = 0\)
• \(\tan 30° = \frac{1}{\sqrt{3}}\approx 0.577\)
• \(\tan 45° = 1\)
• \(\tan 60° = \sqrt{3}\approx 1.732\)
• \(\tan 90°\) is undefined

Understanding these values helps in solving trigonometric equations easily, especially when you're asked to find specific angles related to these values.

Inverse Trigonometric Functions

Inverse trigonometric functions are essential in finding angles when the trigonometric values are known. They are the reverse processes of the original trigonometric functions. The inverse of the tangent function is known as the arctan or \(\tan^{-1}\). This function helps determine an angle when the tangent value is given.

For instance, if you know \(\tan \theta = 1\), you can find \(\theta\) as \(\arctan(1)\). This implies that we are searching for an angle whose tangent is 1. The range of the arctan function is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), or from -90° to 90°, meaning it provides angles within this interval.

Remember, with inverse functions, you're flipping the traditional input-output relationship. Instead of providing an angle to get a side ratio, you give a side ratio to receive an angle. This concept proves incredibly useful in trigonometry and calculus.

Standard Angles

Standard angles are specific angles that have commonly memorized trigonometric function values. These include 0°, 30°, 45°, 60°, and 90°, and extend to angles in all four quadrants like 180°, 270°, and 360°. Each has particular properties that make them core in trigonometry.

When solving a problem like finding \(\arctan(1)\), knowing these angles and their tangent values becomes crucial. At 0° and 90°, tangent values are 0 and undefined, respectively. Meanwhile, \(\tan 45° = 1\); thus, \(\arctan(1) = 45°\).

These standard angles help in simplifying calculations and solving problems without a calculator. By being familiar with their corresponding trigonometric values, you can quickly resolve questions related to trigonometric expressions.