Problem 26 - Technical Mathematics with Calculus

Question

Cotangent, Secant, and Cosecant by Calculator. Evaluate to four decimal places. $$\sec 215.4^{\circ}$$

Step-by-Step Solution

Answer

\(\sec(215.4^\circ)\) rounded to four decimal places.

Step 1: Understand the Secant Function

The secant function is the reciprocal of the cosine function. So to find \(\sec(215.4^\circ)\), you first calculate the cosine of the angle and then take its reciprocal. The secant function is undefined when the cosine is zero, but since the cosine of 215.4 degrees is not zero, we can proceed.

Step 2: Calculate the Cosine of the Angle

Use your calculator to find the cosine of 215.4 degrees. Make sure your calculator is set to the degrees mode. Calculate \(\cos(215.4^\circ)\).

Step 3: Take the Reciprocal of the Cosine Value

Once you have the cosine value, take the reciprocal (which means you divide 1 by the cosine value) to get the secant of the angle. Calculate \(\sec(215.4^\circ) = \frac{1}{\cos(215.4^\circ)}\).

Step 4: Round the Answer

Round the obtained value from the previous step to four decimal places to get the final answer.

Key Concepts

Cosine Function

The cosine function, denoted by cos, is a fundamental component of trigonometry and is related to the right-angled triangle. It is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. When the cosine function is applied to angles in a unit circle, it represents the x-coordinate of the point where the terminal side of the angle intersects the circle.

For any angle in degrees or radians, the value of the cosine function can vary from -1 to 1. To calculate \(\cos(215.4^\circ)\), one needs a calculator or knowledge of cosine values for standard angles and the use of trigonometric identities to find cosine for other specific angles.

Reciprocal Trigonometric Functions

In trigonometry, each basic function has a corresponding reciprocal function. The secant (sec), cosecant (csc), and cotangent (cot) are the reciprocals of the cosine, sine, and tangent functions, respectively. This means that \(\sec(\theta) = \frac{1}{\cos(\theta)}\), \(\csc(\theta) = \frac{1}{\sin(\theta)}\), and \(\cot(\theta) = \frac{1}{\tan(\theta)}\).

These functions are less commonly used than their counterparts but are essential for certain trigonometric calculations and identities. They also have their own graph shapes and properties, such as periods and asymptotes, which can differ significantly from those of the basic functions.

Calculating Secant

To calculate the secant of an angle, we use the relationship between the secant and the cosine functions. Since \(\sec\theta\) is the reciprocal of \(\cos\theta\), you must find the value of the cosine of the angle first. After obtaining \(\cos(215.4^\circ)\) using a calculator, as long as the cosine is not zero, simply take the reciprocal to determine the secant.

Specifically, if \(\cos(215.4^\circ)\) equals a certain value 'X', then \(\sec(215.4^\circ)\) is calculated as \(\frac{1}{X}\). It’s important to keep the angle measurement consistent (degrees or radians) throughout the calculation process. Lastly, the answer should be rounded as required, which often means to a specific number of decimal places for precision.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variables where both sides of the equation are defined. They involve trigonometric functions and are useful in simplifying expressions and solving trigonometric equations. Common identities include the Pythagorean identities, angle sum and difference identities, double angle identities, and reciprocal identities.

Reciprocal identities in particular are crucial when dealing with secant functions. They are the basis for understanding the calculation of secant as performed in the exercise solution. For instance, the reciprocal identity \(\sec(\theta) = \frac{1}{\cos(\theta)}\) is directly used to find the value of the secant function given the cosine of the same angle. Mastery of these identities is fundamental to solving more complex trigonometric problems.