Problem 51 - Precalculus: A Concise Course

Question

Use a calculator to evaluate each function. Round your answers to four decimal places. (Be sure the calculator is in the correct angle mode.) (a) \(\cos 4^{\circ} 50^{\prime} 15^{\prime \prime}\) (b) \(\sec 4^{\circ} 50^{\prime} 15^{\prime \prime}\)

Step-by-Step Solution

Answer

In conclusion, the value of \(\cos 4^{\circ} 50^{\prime} 15^{\prime \prime}\) is approximately 0.9952, and \(\sec 4^{\circ} 50^{\prime} 15^{\prime \prime}\) is approximately 1.0048.

Step 1: Convert the angle from degrees, minutes, and seconds to degrees only

An angle of \(4^{\circ} 50^{\prime} 15^{\prime \prime}\) implies 4 degrees, 50 minutes and 15 seconds. Since one degree is equivalent to 60 minutes, and one minute is equivalent to 60 seconds, we first convert the given time to seconds, and then divide by 3600 (60 minutes/degree * 60 seconds/minute) to convert to degrees. The conversion is as follows: \[4{\text{ degrees }}+ \frac{50{\text{ minutes }}}{60{\text{ minutes/degree }}} + \frac{15{\text{ seconds }}}{3600{\text{ seconds/degree }}} = 4.8375{\text{ degrees }}.\]

Step 2: Calculate the cosine value of the angle

Now that we have the angle in decimal degrees, we can find the cosine of this angle. Ensure the calculator is set to degree mode, then type in the calculated degree, and click the cosine function. Given that \(\cos 4.8375^{\circ}\), when rounded to 4 decimal places, is approximately 0.9952.

Step 3: Calculate the secant of the angle

The secant function is the reciprocal of the cosine function. Therefore, we can find the value of the secant of the angle by taking the reciprocal of the cosine value calculated in Step 2. Since \(\sec 4.8375^{\circ} = \frac{1}{\cos 4.8375^{\circ}}\), the value, when rounded to 4 decimal places is approximately 1.0048.

Key Concepts

Cosine

The cosine function, often denoted as \( \cos \), is one of the fundamental trigonometric functions. It relates the angle in a right triangle to the ratio of the length of the adjacent side over the hypotenuse. In real-world applications, cosine is used to determine distances, model waves, and solve problems involving periodic behavior.
To evaluate the cosine of an angle, especially one given in degrees, minutes, and seconds (like 4° 50' 15"), it is essential to first convert the angle to decimal degrees. This involves understanding that:

• 1 degree = 60 minutes
• 1 minute = 60 seconds

Once converted, the cosine value can easily be calculated using a scientific calculator. Be aware that ensuring your calculator is in degree mode is crucial to avoid inaccuracies. In the given problem, after converting to 4.8375 degrees, the cosine is approximately 0.9952.

Secant

The secant function, denoted as \( \sec \), is closely related to cosine. It is the reciprocal of the cosine function. Mathematically, \( \sec(\theta) = \frac{1}{\cos(\theta)} \). Understanding secant is important because it appears in various mathematical and engineering formulas, especially those related to wave and oscillation equations.
Finding the secant of an angle starts with calculating its cosine. Once the cosine is known, determine the reciprocal to find the secant. For example, if \( \cos(4.8375^{\circ}) \approx 0.9952 \), then:

• \( \sec(4.8375^{\circ}) = \frac{1}{0.9952} \approx 1.0048 \)

This computation allows you to directly understand the relationship between the original angle and its trigonometric presentations.

Angle Conversion

Angle conversion, a fundamental skill in trigonometry, involves changing angles given in degrees, minutes, and seconds into a decimal degree format. This conversion is crucial for performing trigonometric calculations using a calculator since calculators typically use decimal degrees.
The process involves several key steps:

• Convert minutes to degrees: divide the number of minutes by 60.
• Convert seconds to degrees: divide the number of seconds by 3600.

Add these values to the original degree measurement to get the total in decimal degrees.
For example, an angle given as 4° 50' 15" converts as follows:
\[4 + \frac{50}{60} + \frac{15}{3600} = 4.8375\text{ degrees}\]This converted angle can now be used to compute cosine, secant, or other trigonometric values with precision and ease.