Problem 10 - University Calculus: Early Transcendentals

Question

Find the values. $$\sec \left(\cos ^{-1} \frac{1}{2}\right)$$

Step-by-Step Solution

Answer

The value is 2.

Step 1: Understand the Given Expression

We need to evaluate the expression \( \sec \left( \cos^{-1} \frac{1}{2} \right) \). This involves two steps: first, finding the angle whose cosine is \( \frac{1}{2} \), and second, finding the secant of that angle.

Step 2: Determine the Angle for Cosine Inverse

The expression \( \cos^{-1} \frac{1}{2} \) refers to the angle whose cosine value is \( \frac{1}{2} \). We recognize that \( \cos \frac{\pi}{3} = \frac{1}{2} \). Therefore, \( \cos^{-1} \frac{1}{2} = \frac{\pi}{3} \).

Step 3: Find the Secant of the Angle

Secant is the reciprocal of cosine: \( \sec(\theta) = \frac{1}{\cos(\theta)} \). Now, substitute \( \theta = \frac{\pi}{3} \), so \( \sec \left( \frac{\pi}{3} \right) = \frac{1}{\cos \left( \frac{\pi}{3} \right)} = \frac{1}{\frac{1}{2}} = 2 \).

Key Concepts

Understanding the Secant Function

The secant function, denoted as \( \sec(\theta) \), is an important concept in trigonometry. It is the reciprocal of the cosine function.

• Mathematically, this is expressed as \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
• The secant function is one of the six fundamental trigonometric functions, alongside sine, cosine, tangent, cosecant, and cotangent.

Understanding this function helps in calculating values in triangle relationships and circles.
Unlike cosine, which is defined for all real numbers, secant is undefined wherever the cosine equals zero,
as division by zero is not possible.
Secant has a range of \((-\infty, -1] \cup [1, \infty)\), meaning it never falls between \(-1\) and \(1\).
Knowing the secant function’s relation to cosine helps solve various geometric and analytical problems.

Decoding Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find angles when the value of a trigonometric function is given.
In the context of secant functions, understanding \( \cos^{-1} \) or arccos is crucial.

• The expression \( \cos^{-1}(x) \) finds the angle \( \theta \) such that \( \cos(\theta) = x \).
• In our exercise, \( \cos^{-1}(\frac{1}{2}) \) finds the angle whose cosine is \( \frac{1}{2} \).

The principal value of \( \cos^{-1}(\frac{1}{2}) \) is \( \frac{\pi}{3} \), as it satisfies \( \cos(\frac{\pi}{3}) = \frac{1}{2} \).
Inverse trigonometric functions are useful when determining specific angle measures, especially in right triangles
and circles where sine, cosine, and tangent values are already known.
They restrict their outputs to certain intervals, known as principal values, to maintain their inverse nature.

Essentials of Angle Measurement

Angle measurement is a fundamental aspect of trigonometry, allowing us to compare and calculate angles in various systems.
Angles are most commonly measured in degrees or radians, with each offering unique advantages in calculations.

• One full rotation around a circle equals \(360\) degrees or \(2\pi\) radians.
• The radian is a standard unit in calculus and physics because of its natural mathematical properties.
  In trigonometry, angles are frequently expressed in radians for simplicity, especially in formulas and computations.

Understanding angle measurement is crucial when interpreting functions like \( \sec(\theta) \)\( \cos^{-1} \). When someone refers to \( \cos^{-1}(\frac{1}{2}) \), knowing it equates to \( \frac{\pi}{3} \) radians is key.
Hence, proficiency in converting between degrees and radians provides an edge in understanding and applying trigonometric concepts effectively.