Problem 117 - Precalculus

Question

For the following exercises, evaluate the expressions. $$ \tan ^{-1}(1) $$

Step-by-Step Solution

Answer

\( \tan^{-1}(1) = \frac{\pi}{4} \)

Step 1: Understanding Inverse Trigonometric Function

The function we are dealing with is \( \tan^{-1}(1) \), which is an inverse trigonometric function. \( \tan^{-1}(x) \) or \( \arctan(x) \) gives us the angle \( \theta \) such that \( \tan(\theta) = x \). So, we need to find an angle \( \theta \) where \( \tan(\theta) = 1 \).

Step 2: Identifying the Angle for Tangent of 1

Recall the unit circle or your knowledge of special angles. We know that \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) and that \( \tan(\theta) = 1 \) when the sine and cosine values are equal. During one full circle, this occurs at \( \theta = \frac{\pi}{4} \) or \( 45^\circ \), where both sine and cosine are \( \frac{\sqrt{2}}{2} \).

Step 3: Range of Inverse Tangent Function

The range of the inverse tangent function \( \tan^{-1}(x) \) is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \). Therefore, the angle that satisfies \( \tan(\theta) = 1 \) is \( \theta = \frac{\pi}{4} \), since it lies within the specified range.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental relationships in trigonometry that involve the six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. These identities allow us to simplify trigonometric expressions and convert one trigonometric function into another.

Some essential identities include:

• Reciprocal Identities: These express functions like sine and cosine in terms of their reciprocals. For example, \( \sin(\theta) = \frac{1}{\csc(\theta)} \).
• Pythagorean Identities: These relate the squares of sine and cosine to 1, such as \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
• Ratio Identities: These express the tangent function as a ratio of sine and cosine: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).

These identities are used to solve problems and prove other trigonometric statements, making them indispensable tools in trigonometry. In the context of inverse functions, knowing that \( \tan(\theta) = 1 \) is crucial, particularly since it indicates angles where sine and cosine are equal.

Special Angles

Special angles are those angles that are frequently used in trigonometry because they produce easily recognizable and beautifully simple values for sine, cosine, and tangent functions.

These angles include:

• \(0\degree, 30\degree, 45\degree, 60\degree, \text{and}\ 90\degree \)
• Their radian equivalents: \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} \)

For example, at \(45\degree\) or \(\frac{\pi}{4}\), both sine and cosine equal \(\frac{\sqrt{2}}{2}\). Thus, \(\tan(\frac{\pi}{4}) = 1\) because \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\). Recognizing these angles quickly simplifies evaluating inverse trigonometric functions, like the one in our original problem, where identifying \(\tan^{-1}(1)\) leads straight to \(\frac{\pi}{4}\).