Problem 2 - Essential Calculus Early Transcendentals

Question

Find the exact value of each expression. (a) \(\tan ^{-1}(1 / \sqrt{3})\) (b) \(\sec ^{-1} 2\)

Step-by-Step Solution

Answer

(a) \( \frac{\pi}{6} \) radians; (b) \( \frac{\pi}{3} \) radians.

Step 1: Understand the Problem

The problem asks us to find the exact values of two inverse trigonometric functions: (a) the inverse tangent of \( \frac{1}{\sqrt{3}} \) and (b) the inverse secant of 2. Let's evaluate each part separately.

Step 1: Solve for (a)

We need to find the angle \( \theta \) such that \( \tan(\theta) = \frac{1}{\sqrt{3}} \). Recall that \( \tan(30^\circ) = \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \). Thus, \( \tan^{-1}(\frac{1}{\sqrt{3}}) = \frac{\pi}{6} \) radians.

Step 2: Solve for (b)

We need to find the angle \( \theta \) such that \( \sec(\theta) = 2 \). Recall that \( \sec(\theta) = \frac{1}{\cos(\theta)} \), so \( \cos(\theta) = \frac{1}{2} \). The angle \( \theta \) that satisfies \( \cos(\theta) = \frac{1}{2} \) is \( \theta = \frac{\pi}{3} \). Thus, \( \sec^{-1}(2) = \frac{\pi}{3} \) radians.

Step 4: Conclusion

The exact value of \( \tan^{-1}(\frac{1}{\sqrt{3}}) \) is \( \frac{\pi}{6} \) and the exact value of \( \sec^{-1}(2) \) is \( \frac{\pi}{3} \).

Key Concepts

Inverse Tangent

The inverse tangent function, often denoted as \( \tan^{-1} \) or \( \arctan \), is crucial in determining angles when given a tangent value. Essentially, this function answers the question: "What angle has this particular tangent value?" For example, if you are asked to find \( \tan^{-1}(1/\sqrt{3}) \), you're identifying an angle with a tangent of \( 1/\sqrt{3} \).
To solve this, recall the basic trigonometric angles. One such angle is 30 degrees, or \( \frac{\pi}{6} \) radians, where \( \tan \left( \frac{\pi}{6} \right) = \frac{1}{\sqrt{3}} \).

• When solving \( \tan^{-1} \), it’s helpful to remember trigonometric values for common angles like 30°, 45°, and 60°.
• Inverse tangent values lie between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), which is the range of its principal value.

Understanding \( \tan^{-1} \) helps solve many geometry and physics problems involving angles.

Inverse Secant

The inverse secant, represented as \( \sec^{-1} \), is a way to find angles if you know their secant value. Secant is the reciprocal of cosine, meaning \( \sec(\theta) = \frac{1}{\cos(\theta)} \). For the exercise of finding \( \sec^{-1}(2) \), consider that you're seeking an angle with a secant of 2.
This leads us to the equation \( \cos(\theta) = \frac{1}{2} \). Through fundamental angle knowledge, we recognize this cosine value corresponds to \( \theta = \frac{\pi}{3} \).

• In practice, \( \sec^{-1} \) values are typically within \( [0, \pi] \), excluding \( \frac{\pi}{2} \).
• While working with inverse secants, identifying cosine values for common angles helps find exact solutions swiftly.

Using inverse secant efficiently requires familiarity with its domain and the reciprocal trigonometric relationships.

Exact Trigonometric Values

Understanding exact trigonometric values is vital for solving problems involving inverse trigonometric functions. These values arise from commonly used angles, often found in geometry and calculus. Knowing these exact values allows quick evaluation of trigonometric expressions.
For instance, angles like \( 30^\circ \), \( 45^\circ \), and \( 60^\circ \) correspond to known trigonometric values:

• \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \)
• \( \cos(60^\circ) = \frac{1}{2} \)
• \( \sec(60^\circ) = 2 \)

By understanding and memorizing these values, you can solve inverse problems efficiently. The ability to use these values reduces cognitive load during problem-solving, making calculations more intuitive.