Problem 21 - Technical Mathematics with Calculus

Question

Reciprocal Relationships. Evaluate to four decimal places. Find \(\sec \theta\) if \(\cos \theta=0.7354\)

Step-by-Step Solution

Answer

\(\sec(\theta) \approx 1.3598\) to four decimal places.

Step 1: Understanding the Relationship Between Secant and Cosine

The secant function, denoted as \(\sec\), is the reciprocal of the cosine function. This means that \(\sec(\theta) = \frac{1}{\cos(\theta)}\). To find \(\sec(\theta)\) when provided with \(\cos(\theta)\), simply take the reciprocal of \(\cos(\theta)\).

Step 2: Finding the Reciprocal of Cosine

Since \(\cos(\theta) = 0.7354\), we calculate the reciprocal to determine \(\sec(\theta)\). Perform the division: \(\sec(\theta) = \frac{1}{0.7354}\).

Step 3: Calculating and Rounding

Using a calculator, divide 1 by 0.7354 to get \(\sec(\theta)\) to four decimal places. Ensure that the result is rounded correctly to four decimal places as required.

Key Concepts

Reciprocal Relationships in Trigonometry

In trigonometry, understanding reciprocal relationships is fundamental. A reciprocal relationship in mathematics refers to a pair of numbers that multiply to one. For trigonometric functions, each function has a reciprocal: sine (sin) has cosecant (csc), cosine (cos) has secant (sec), and tangent (tan) has cotangent (cot). Specifically, the secant function is the reciprocal of the cosine function, which can be expressed as
\[ \sec(\theta) = \frac{1}{\cos(\theta)} \].

Cosine Function

The cosine function, denoted as \(\cos\), is a basic trigonometric function that represents the ratio of the adjacent side to the hypotenuse of a right-angled triangle. It's important to know the cosine function because it is widely used not only in trigonometry but also in calculus, physics, and engineering. When you have the value of the cosine of an angle, this value can help you find the lengths of sides in a triangle or analyze periodic phenomena like waves.

Calculating Secant Values

To calculate the value of secant, you simply need to find the reciprocal of the cosine value. The secant function can be tricky because it is not as frequently used as the sine or cosine, and it may not have a button on your calculator. Calculating secant values involves taking the reciprocal of a given cosine value; for example, if \(\cos(\theta) = 0.7354\), then \(\sec(\theta) = \frac{1}{0.7354}\). It is a straightforward division, but remember, since secant is the reciprocal, you will get a value greater than 1 whenever the cosine value is less than 1.

Division and Rounding in Mathematics

Division is one of the arithmetic operations and is used to describe the fair sharing or grouping of a number. It is the process of calculating the amount of times one number is contained within another. When working with division in mathematical problems, especially with irrational numbers or when seeking precision to a certain number of decimal places, rounding can be important. Rounding is the process of adjusting the digits of a number to make it simpler but still close in value to the original number. For example, when calculating \(\sec(\theta)\), after dividing 1 by the cosine value, we round the result to four decimal places for precision and simplicity in further calculations.