Question

Evaluating inverse trigonometric functions Without using a calculator, evaluate or simplify the following expressions. $$\tan \left(\tan ^{-1} 1\right)$$

Short answer

Answer: The value of the expression \(\tan (\tan^{-1} 1)\) is 1.

Step-by-step solution

Understand the inverse function

In this expression, \(\tan^{-1} 1\) represents the angle whose tangent is 1.

Finding the angle

We know that tangent function has properties such as: $$\tan\left(\frac{\pi}{4}\right) = 1$$ So, we can find the angle $$\tan^{-1} 1 = \frac{\pi}{4}$$

Calculate tangent of the angle

Now, we need to calculate the tangent of the angle we have found: $$\tan \left(\frac{\pi}{4}\right)$$

Evaluate the expression

Since we know that $$\tan\left(\frac{\pi}{4}\right) = 1$$ Therefore, the expression becomes: $$\tan \left(\tan ^{-1} 1\right) = \tan \left(\frac{\pi}{4}\right) = 1$$ The simplified expression is equal to 1.

Key concepts

Understanding Trigonometric Identities

Trigonometric identities are equations that relate the angles and ratios of a right triangle. They are extremely useful when working with trigonometric functions like sine, cosine, and tangent. These identities help simplify complex trigonometric expressions and solve equations involving these functions.
Some basic trigonometric identities include:

• Pythagorean Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Tangent Identity: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
• Reciprocal Identities, such as \( \csc \theta = \frac{1}{\sin \theta} \)

Using trigonometric identities, we can simplify expressions, like recognizing that \( \tan(\frac{\pi}{4}) = 1 \). This specific identity is especially useful in problems involving the tangent function and its inverse.

Exploring the Tangent Function

The tangent function, often noted as \( \tan \theta \), is one of the fundamental trigonometric functions. It represents the ratio of the opposite side to the adjacent side in a right triangle. Essential properties of the tangent function include:

• Periodicity: The tangent function repeats every \( \pi \) radians, making it a periodic function.
• Vertical Asymptotes: Tan has undefined values (vertical asymptotes) at \( \theta = \frac{\pi}{2} + n\pi \), where \( n \) is an integer.
• Symmetry: It is an odd function since \( \tan(-\theta) = -\tan(\theta) \).

A crucial property is that \( \tan \left( \frac{\pi}{4} \right) = 1 \). This property is used frequently when solving expressions with the inverse tangent function, \( \tan^{-1} \).

Demystifying Inverse Functions

Inverse trigonometric functions allow us to determine angles from known trigonometric ratios. For the tangent function, the inverse is written as \( \tan^{-1} \) or \( \arctan \). These functions reverse the process of the initial basic trigonometric functions:

• \( \tan^{-1}(x) \) gives the angle \( \theta \) such that \( \tan(\theta) = x \).
• It has a range of \( (-\frac{\pi}{2}, \frac{\pi}{2}) \) due to its continuous, non-periodic nature over this interval.

Understanding the inverse function provides clarity in expressions like \( \tan(\tan^{-1}(1)) \). When you see \( \tan^{-1}(1) \), it's an invitation to find the angle whose tangent equals 1, which we know to be \( \frac{\pi}{4} \). Subsequently, applying the tangent function to this angle retrieves the original ratio, thus simplifying the expression back to 1.