Problem 81 - A Graphical Approach to Precalculus with Limits

Question

Give the exact real number value of each expression. Do not use a calculator. $$\cos \left(\tan ^{-1}(-2)\right)$$

Step-by-Step Solution

Answer

\( \cos(\tan ^{-1}(-2)) = \frac{\sqrt{5}}{5} \)

Step 1: Define the Inverse Tangent Function

The expression involves \( \tan^{-1}(-2) \). This means we are looking for an angle \( \theta \) such that \( \tan(\theta) = -2 \). We need to find \( \cos(\theta) \) when \( \tan(\theta) = -2 \).

Step 2: Use the Pythagorean Identity

For \( \tan(\theta) = -2 \), by the Pythagorean identity, \( 1 + \tan^2(\theta) = \sec^2(\theta) \). Substitute \( \tan(\theta) = -2 \):\[1 + (-2)^2 = \sec^2(\theta)\]which simplifies to:\[1 + 4 = \sec^2(\theta)\]\[\sec^2(\theta) = 5\].

Step 3: Calculate \( \cos(\theta) \) from \( \sec(\theta) \)

Since \( \sec^2(\theta) = 5 \), it follows that \( \sec(\theta) = \sqrt{5} \) or \( \sec(\theta) = -\sqrt{5} \). Because \( \tan(\theta) = -2 \), and considering the range of \( \tan^{-1} \), \( \theta \) is in the fourth quadrant where cosine is positive. Thus, \( \sec(\theta) = \sqrt{5} \). Therefore, \( \cos(\theta) = \frac{1}{\sqrt{5}} \).

Step 4: Rationalize the Cosine Value

We have \( \cos(\theta) = \frac{1}{\sqrt{5}} \). To rationalize, multiply numerator and denominator by \( \sqrt{5} \):\[\cos(\theta) = \frac{1 \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} = \frac{\sqrt{5}}{5}\].

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are crucial for finding angles when given certain trigonometric ratios. In this exercise, the focus is on the inverse tangent function, denoted as \( \tan^{-1} \).

• \( \tan^{-1}(x) \) is used to determine an angle \( \theta \) such that \( \tan(\theta) = x \).
• In the context of this problem, we deal with \( \tan^{-1}(-2) \), meaning we seek the angle whose tangent value equals \(-2\).
• The value for \( \tan^{-1} \) always results in an angle within a specific range. For tangent, this is usually \(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\) (or \(-90^{\circ} < \theta < 90^{\circ}\))—the angles where tangent is defined and one-to-one.

Understanding inverse trigonometric functions helps to unravel how angles interact with their respective trigonometric ratios. Once you have the angle, further computations, such as finding sine or cosine, become straightforward using identities and relationships.

Pythagorean Identity

The Pythagorean identity is an essential concept for connecting different trigonometric functions, most notably sine, cosine, and tangent.

• The identity states that \( 1 + \tan^2(\theta) = \sec^2(\theta) \), showcasing a vital relationship between tangent and secant.
• In our particular example, knowing \( \tan(\theta) = -2 \), we substitute \( \tan^2(\theta) = 4 \) into the identity: \( 1 + 4 = \sec^2(\theta) \).
• This leads to \( \sec^2(\theta) = 5 \), from which we deduce that \( \sec(\theta) = \pm \sqrt{5} \).

Choosing the correct sign depends on the quadrant where \( \theta \) falls, heavily influencing the sign of secant (and by extension cosine). In the fourth quadrant, where \( \tan^{-1}(-2) \) places \( \theta \), cosine is positive, confirmed by selecting \( \sec(\theta) = \sqrt{5} \). This aligns with understanding how each trigonometric function behaves in different quadrants.

Angle Calculations

Calculating angles from given trigonometric functions without a calculator relies on identities and understanding the function's characteristics.

• First, identify the known ratio (e.g., \( \tan(\theta) = -2 \)), using it to find \( \theta \) in the appropriate range.
• Next, apply relevant identities to find other functions, like the Pythagorean identity to link tangent and secant in this exercise.
• Consider the quadrant to determine the sign of these functions. For \( \theta \) in this problem, being in the fourth quadrant ensures a positive cosine value.

Once secant is found, inverse relationships provide \( \cos(\theta) \). Since \( \sec(\theta) = \frac{1}{\cos(\theta)} \), from \( \sec(\theta) = \sqrt{5} \), we calculate:\( \cos(\theta) = \frac{1}{\sqrt{5}} \) and further rationalization yields \( \frac{\sqrt{5}}{5} \). This process highlights the integration of identities and inverse functions to precisely calculate angles.