Problem 57 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of inverse functions to find the exact value of the expression, if possible. \(\tan \left[\tan ^{-1}(-0.5)\right]\)

Step-by-Step Solution

Answer

The exact value of the expression \(\tan \left[ \tan^{-1}(-0.5) \right]\) is -0.5.

Step 1: Identify original value

In the expression \( \tan\left[\tan^{-1}(-0.5)\right] \), identify \( -0.5 \) as the original value, denoted as \( x \).

Step 2: Apply properties of inverse functions

Use the property of inverse functions to simplify the expression. The inverse function essentially reverses the original function. If \( f^{-1}(f(x)) = x \) is true, then \( \tan(\tan^{-1}(x)) = x \) is also true. Applying this property to the given expression, \( \tan(\tan^{-1}(-0.5)) = -0.5 \).

Step 3: Form a conclusion

So \( \tan\left[\tan^{-1}(-0.5)\right] \) simplifies to -0.5, which is the exact value of the expression.

Key Concepts

Tangent Function

The tangent function, often written as \( \tan(x) \), is one of the primary trigonometric functions. It is defined as the ratio of the sine and cosine values of a given angle \( x \). In mathematical terms, this can be expressed as:

• \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)

This function is periodic, meaning it repeats its values in regular intervals. For the tangent function, this period is \( \pi \) radians (or 180 degrees). You will notice that the tangent function has vertical asymptotes at certain points, specifically where the cosine value is zero because we cannot divide by zero. At angles like \( \frac{\pi}{2} + k\pi \) (where \( k \) is an integer), the cosine is zero, and the tangent is undefined.
Understanding the tangent function is crucial, especially when dealing with its inverse, as seen in the problem \( \tan[\tan^{-1}(-0.5)] \). Here, recognizing that \(-0.5\) is a ratio of sine to cosine helps simplify the expression, as the tangent and its inverse cancel each other out under specific conditions.

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverse operations of the trigonometric functions. These allow you to find an angle when you know the function's value. For example, the inverse tangent function is written as \( \tan^{-1}(x) \) or \( \arctan(x) \). It takes a ratio as input and outputs the corresponding angle whose tangent is that ratio.

• The principal value range for \( \tan^{-1}(x) \) is \(-\frac{\pi}{2} < y < \frac{\pi}{2} \).
• This function is designed to undo what the tangent function does, meaning \( \tan(\tan^{-1}(x)) = x \), provided \( x \) is within its codomain.

This property is pivotal when solving problems that involve simplifying expressions like \( \tan[\tan^{-1}(-0.5)] \). Here, it directly applies this concept by stating that the inverse function and the original function cancel each other's operations within the specified range, yielding the original number, \(-0.5\), in this case.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the involved variables. They provide relationships between different trigonometric functions. Identities like Pythagorean, addition, and product-to-sum are frequently used to solve trigonometric equations or simplify expressions.

• One basic identity involving tangent is: \( \tan^2(x) + 1 = \sec^2(x) \)
• Another useful identity is the angle sum identity: \( \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} \)

In this context of inverse trigonometric functions, the identity \( \tan(\tan^{-1}(x)) = x \) acts like an equation rather than a transformation of \( x \), when \( x \) is in the correct range.
Utilizing these identities helps not only in understanding fundamental trigonometric properties but also in applying them correctly when simplifying expressions or solving more complex problems. They serve as the foundation of trigonometry and are essential for calculations involving angles and sides within right-angled triangles.