Problem 87 - Precalculus

Question

Find the smallest positive measure of \(\theta\) (rounded to the nearest degree) if the indicated information is true. \(\sec \theta=1.0001\) and the terminal side of \(\theta\) lies in quadrant I.

Step-by-Step Solution

Answer

The smallest positive measure of \( \theta \) is approximately \( 1^{\circ} \).

Step 1: Understand the relationship between secant and cosine

The secant function is the reciprocal of the cosine function. Therefore, if \( \sec \theta = 1.0001 \), then \( \cos \theta = \frac{1}{\sec \theta} = \frac{1}{1.0001} \approx 0.9999 \).

Step 2: Determine the angle \theta\) from \cos \theta\)

To find \( \theta \), we use the inverse cosine function: \( \theta = \cos^{-1}(0.9999) \). This is because \( \cos \theta \approx 0.9999 \), and we need to find the angle whose cosine value is this number.

Step 3: Calculate the angle \theta\)

Using a calculator, compute \( \theta = \cos^{-1}(0.9999) \). The inverse cosine of 0.9999 is approximately \( 1.00^{\circ} \).

Step 4: Verify the quadrant

The problem states the terminal side of \( \theta \) lies in quadrant I. Since angles in quadrant I range from \( 0^{\circ} \) to \( 90^{\circ} \), our solution of \( 1^{\circ} \) fits because it is the smallest positive measure in quadrant I.

Key Concepts

Secant Function

The secant function, often abbreviated as "sec," is one of the six fundamental trigonometric functions. It is particularly important because it is defined as the reciprocal of the cosine function. Mathematically, this relationship is expressed as:

• \( \sec \theta = \frac{1}{\cos \theta} \)

This means that if you know the secant of an angle, you can easily determine its cosine, and vice versa. In this exercise, we have \( \sec \theta = 1.0001 \), which implies:

• \( \cos \theta = \frac{1}{1.0001} \approx 0.9999 \)

The secant function can be particularly useful when angle measures and distances on the unit circle are involved. It helps to identify the opposite and adjacent sides of a right triangle, aiding in broader trigonometric problem-solving. Remember, understanding the reciprocal relationship is key, as small changes in secant values can indicate angle measures close to the horizon in a unit circle perspective.

Cosine Function

The cosine function, denoted as \( \cos \theta \), is another critical trigonometric function. It represents the x-coordinate of the point on the unit circle corresponding to an angle \( \theta \). For any given angle, the cosine function helps us understand how much the terminal side of this angle stretches along the x-axis.

• In this exercise, we've derived that \( \cos \theta = 0.9999 \), meaning that the angle \( \theta \) is almost horizontal.
• The function values range from -1 to 1, where 1 indicates the angle's terminal side completely aligns with the positive x-axis.

Knowing the cosine value is paramount in determining angles, lengths, and utilizing the Pythagorean theorem in equations involving triangles. It connects directly with functions like sine and tangent, forming foundational trigonometric identities. In this particular exercise, the high cosine value is indicative of a very small angle, as seen when \( \cos \theta \approx 0.9999 \) yields an angle measure of about 1 degree, just slightly deviating from the x-axis. This proximity to 1 reflects minimal change in angle, emphasizing linearity along the x-axis.

Inverse Trigonometric Functions

Inverse trigonometric functions are designed to retrieve angle measures from known trigonometric values. The particular function we used here is the inverse cosine, denoted as \( \cos^{-1} \). This function helps us find the angle \( \theta \) when we know its cosine value. The relationship is expressed as follows:

• If \( \cos \theta = 0.9999 \), then \( \theta = \cos^{-1}(0.9999) \).
• This calculation provides us with the smallest angle in degrees for which the cosine equals 0.9999.

This tool is incredibly valuable when dealing with angle measures that aren't immediately apparent, allowing for conversion of trigonometric values back into angles. In the context of this problem, we leverage \( \cos^{-1} \) to conclude that the angle \( \theta = 1.00^{\circ} \), effectively verifying it fits the criteria of lying in the first quadrant, where angles range between 0 and 90 degrees. Understanding how inverse functions operate gives insight into reversing processes of trigonometric calculations, crucial for solving problems involving reciprocal trigonometric functions.