Problem 50 - 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) \(\cot 79.56^{\circ}\) (b) \(\sec 79.56^{\circ}\)

Step-by-Step Solution

Answer

The obtained values (after rounding off to 4 decimal places) are: \[\cot 79.56^{\circ}\] = -0.0175, and \[\sec 79.56^{\circ}\] = 5.7580

Step 1: Setup the Calculator

First, the calculator must be set to degree mode since the given angles are in degrees. This setting is often found in the 'Mode' or 'Settings' section of many scientific calculators.

Step 2: Evaluate cotangent function

Input 79.56 into the calculator, then look for the 'cot' button. Press it to find the cotangent of the given angle. The calculator will provide the operation's result. Now round to four decimal places as per the question's requirements.

Step 3: Evaluate secant function

Repeat the process similar to Step 2, but this time look for the 'sec' button. After obtaining the secant of 79.56, round it to four decimal places.

Key Concepts

Cotangent Function

When diving into the realm of trigonometry, the cotangent function, often denoted as \(\cot\), emerges as a fundamental ratio in right triangle geometry and in the unit circle. It is defined as the ratio of the adjacent side to the opposite side of a right triangle, or more commonly in trigonometry, as the reciprocal of the tangent function: \(\cot(\theta) = \frac{1}{\tan(\theta)}\).

Understanding the cotangent function can be facilitated by considering it within the context of the unit circle, where it represents the coordinate of a point that lies on the circle at a specific angle \(\theta\). In terms of the exercise you're working on, evaluating \(\cot(79.56^\circ)\) would be the same as looking for the reciprocal of the tangent of an angle of approximately 79.56 degrees.

It's crucial to note that the cotangent function can exhibit very large values or approach zero depending on the angle's measure. That's why rounding off to four decimal places as stipulated in the exercise is not just a matter of precision, but also of practical interpretation, allowing for easier grasp and application of the trigonometric ratios in various scenarios.

Secant Function

Diving into another core concept, the secant function plays a pivotal role in trigonometry. Symbolized as \(\sec\), the secant is defined as the reciprocal of the cosine function: \(\sec(\theta) = \frac{1}{\cos(\theta)}\). Think of the cosine function as describing the horizontal coordinate of a point on the unit circle corresponding to a given angle \(\theta\); thereby, the secant gives a measure of the hypotenuse when the adjacent side length is one in a right-angled triangle.

Key to realizing is that the secant function can also take on values greater than one or less than negative one, as it is the ratio of the length of the hypotenuse over the adjacent side in the right triangle representation. As the adjacent side becomes smaller, the secant value can increase significantly. When you were asked to find \(\sec(79.56^\circ)\), rounding the result to four decimal places ensures the output is practical for use in further calculations or modeling.

Calculator Degree Mode

The final key aspect to navigate effectively in trigonometric evaluations involves the angle measurement settings of your calculator - the calculator degree mode. Calculators can typically operate in both degree and radian modes, and making certain your calculator is in the correct mode is essential for accurate computations.

In the context of your exercise, specifying that the angles are given in degrees \(79.56^\circ\) highlights the need to use the degree mode prior to any calculations. This mode will ensure that when you use functions like cotangent or secant, the calculator interprets the input angle as degrees, not radians. To set a calculator to degree mode, you will often navigate to a 'Mode' or 'Settings' menu and select the degree option. Failing to do so could lead to incorrect results, as the calculations would be conducted as though the input were radian measures - a common and avoidable error.

Always double-check the calculator's angle setting before starting your math problems to circumvent potential mismatches in answers, especially when transitioning between exercises that may require different modes.