Problem 74 - A Graphical Approach to Precalculus with Limits

Question

Give the exact real number value of each expression. Do not use a calculator. $$\tan \left(\tan ^{-1}(-4)\right)$$

Step-by-Step Solution

Answer

The expression evaluates to -4.

Step 1: Understand the Inverse Function

The inverse tangent function, denoted as \( \tan^{-1}(x) \), gives the angle \( \theta \) such that \( \tan(\theta) = x \).

Step 2: Apply the Inverse Tangent Function

Given \( \tan^{-1}(-4) \), we need to find an angle \( \theta \) such that \( \tan(\theta) = -4 \). The angle \( \theta \) is in the range \( (-\frac{\pi}{2}, \frac{\pi}{2}) \).

Step 3: Use the Property of the Tangent Function

Given that \( \theta = \tan^{-1}(-4) \), substituting this back into the tangent function gives \( \tan(\theta) = -4 \).

Step 4: Evaluate the Original Expression

Now evaluate \( \tan(\tan^{-1}(-4)) \). Since the tangent and inverse tangent functions are inverse operations, they cancel each other out, resulting in the original argument \( -4 \).

Key Concepts

Inverse Functions

Inverse functions are a fundamental concept in mathematics, especially when dealing with trigonometric functions. An inverse function essentially 'reverses' the effect of the original function. For instance, if you start with a value, apply a function, and then apply its inverse function, you'll end up with the starting value again.

In the case of trigonometric functions, the inverse functions are denoted by adding a "-1" exponent. For example, the inverse tangent is written as \( \tan^{-1}(x) \) and is also known as the arctangent function.

• This function takes a real number and returns an angle.
• The angle is such that the original tangent of this angle equals the given number.

When you encounter an expression like \( \tan(\tan^{-1}(x)) \), the tangent and its inverse essentially "cancel out," leaving you with \( x \). This property is significant because it simplifies seemingly complex trigonometric expressions quickly.

Tangent Function

The tangent function is one of the three primary trigonometric functions, alongside sine and cosine. It is often thought of in terms of a right triangle, where the tangent of an angle is the ratio of the length of the opposite side to the adjacent side.

Expressed mathematically, for an angle \( \theta \), it is denoted as \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).

• The tangent function is repetitive, and its values cycle every \( \pi \) radians or 180 degrees.
• Unlike sine and cosine, the tangent function can take any real number value.

The range of \( \tan^{-1}(x) \), or arctangent, is restricted to \( (-\frac{\pi}{2}, \frac{\pi}{2}) \). This limitation is crucial when solving problems involving inverse trigonometric functions, such as in the original exercise. By understanding this range, you ensure that any inverse tangent problem results in a valid angle.

Angle Measurement

Angles can be measured in different units, but the two most common are degrees and radians. In trigonometry, radians are often preferred because they provide a more natural mathematical relationship between the angle's length (arc) and the radius of the circle.

Radians relate directly to the properties of circles. The circumference of a circle is \(2\pi\) times the radius, meaning one full rotation is \(2\pi\) radians. Therefore, the angle in radians is a measure of the distance along the circle's edge compared to its radius.

• When converting between degrees and radians, remember that \(180\) degrees is equal to \(\pi\) radians.
• An angle of \(90\) degrees is equivalent to \(\frac{\pi}{2}\) radians.

In the context of inverse trigonometric functions, such as \( \tan^{-1}(x) \), the resulting angle is usually in radians to keep calculations simple and precise. Understanding this is key to grappling with trigonometric problems, particularly where the range of the inverse function is considered, as in the exercise given.