Problem 31 - Precalculus: Functions and Graphs

Question

Approximate the acute angle \(\theta\) to the nearest (a) \(0.01^{\circ}\) and (b) \(1^{\prime}\) $$\sec \theta=4.246$$

Step-by-Step Solution

Answer

(a) \(76.38^{\circ}\) (b) \(76^{\circ} 23'\)

Step 1: Understand the relationship between secant and cosine

Recall that the secant function is the reciprocal of the cosine function. Thus, we can express \( \sec \theta \) in terms of \( \cos \theta \) as: \( \sec \theta = \frac{1}{\cos \theta} \).

Step 2: Calculate \( \cos \theta \)

Since \( \sec \theta = 4.246 \), we can find \( \cos \theta \) by taking the reciprocal: \( \cos \theta = \frac{1}{4.246} \). Compute this value to find \( \cos \theta \approx 0.2355 \).

Step 3: Find \( \theta \) using arccosine

Use the inverse cosine function to find \( \theta \). Calculate \( \theta = \cos^{-1}(0.2355) \). Upon computation, \( \theta \approx 76.3844^{\circ} \).

Step 4: Round \( \theta \) to the nearest 0.01°

Round the value from Step 3 to the nearest hundredth of a degree. \( \theta = 76.38^{\circ} \).

Step 5: Convert and round \( \theta \) to minutes

First, separate the whole number from the fractional degree: \( 76^{\circ} \) and \( 0.3844^{\circ} \). Convert \( 0.3844^{\circ} \) to minutes by multiplying by 60 (since 1° = 60 minutes): \( 0.3844 \times 60 \approx 23.064 \). Round to the nearest minute: \( 23' \). Thus, \( \theta \approx 76^{\circ} 23'\).

Key Concepts

Secant Function

The secant function, often denoted as \( \sec \theta \), is a key concept in trigonometry, intimately connected to the cosine function. In fact, the secant function is the reciprocal of the cosine function. This means that for any angle \( \theta \), the value of the secant is found by taking one divided by the cosine of that angle.
Think of it like this: if \( \cos \theta = \frac{3}{5} \), then \( \sec \theta = \frac{5}{3} \).
It's important because the secant function, like other trigonometric functions, helps in relating different angles and sides of a triangle, especially in non-right triangles. Here are some handy tips:

• Since the cosine value can range between -1 and 1, the secant function takes on values from positive or negative infinity up to -1, and from 1 upward to positive infinity. It never touches the range in between.
• In terms of a graph, secant can have vertical asymptotes wherever cosine is zero because dividing by zero is undefined.

Knowing this can help understand problems like calculating an angle when you know its secant value, as illustrated in the original problem.

Cosine Function

The cosine function is one of the primary trigonometric functions and is fundamental in the study of angles and triangles. It is often denoted by \( \cos \theta \). For an angle \( \theta \), the cosine represents the ratio of the adjacent side to the hypotenuse in a right triangle.
Mathematically, this can be described as: \[ \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
This function is especially useful because it allows you to determine unknown side lengths of a triangle if one side and one angle are known.

• The cosine function outputs values between -1 and 1. This bounded nature is crucial for understanding the behavior of cosine in various applications, ranging from solving geometry problems to modeling periodic phenomena.
• The cosine graph is a wave that begins at its maximum of 1 when \( \theta = 0 \) and oscillates between 1 and -1.

The connection between cosine and its reciprocal, the secant function, aids in exploring deeper trigonometric relationships, like solving angles when the secant is known.

Inverse Trigonometric Functions

Inverse trigonometric functions are powerful tools used to retrieve angles from known trigonometric values. For example, the inverse cosine function, denoted as \( \cos^{-1} \), helps find the angle \( \theta \) when the cosine of \( \theta \) is given.
When you see an expression like \( \theta = \cos^{-1}(0.2355) \), you're asking what angle has 0.2355 as its cosine. Inverse functions in trigonometry allow us to move backward from a ratio back to the angle itself. Here are some fundamental points:

• The range of \( \cos^{-1} \) is from 0 to 180 degrees, meaning it only provides solutions in this interval because cosine itself is not a one-to-one function outside this range.
• Knowing the inverse functions aids in fields such as physics and engineering, where precise angle calculations are necessary.

In the original problem, inverse cosine was used to determine angle \( \theta \), demonstrating how these functions bridge numerical values and their geometrical interpretations.