Problem 33 - Algebra 2

Question

Use the graph of the appropriate reciprocal trigonometric function to find each value. Round to four decimal places. $$ \sec 30^{\circ} $$

Step-by-Step Solution

Answer

The rounded value of \(\sec 30^{\circ}\) is approximately \(1.1557\).

Step 1: Identify the Trigonometric Reciprocal

In this step, recall that the secant \(\sec\) of an angle is the reciprocal of the cosine \(\cos\) of that angle. Therefore, we can rewrite \(\sec 30^{\circ}\) as \(1/\cos 30^{\circ}\).

Step 2: Compute the Cosine

Next, remember the known value of the cosine of 30 degree. Using the given angle, compute \(\cos 30^{\circ}\). The value of \(\cos 30^{\circ}\) in decimal form is \(0.8660\).

Step 3: Find the Reciprocal

The secant of the given angle is the reciprocal of the cosine of that angle. To compute the secant, take the reciprocal of the cosine computed in the previous step. So, \(1/0.8660 = 1.1557\). After performing the division, it's found that \(\sec 30^{\circ} \approx 1.1557\).

Step 4: Round the Secant Value

As the last part of this procedure, round off the computed secant value to four decimal places as indicated by the exercise. Thus, after rounding, \(\sec 30^{\circ} \approx 1.1557\).

Key Concepts

Secant

The secant function, denoted as \( \sec \theta \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the cosine function. Therefore, for an angle \( \theta \), the secant is written as \( \sec \theta = \frac{1}{\cos \theta} \).
When working with angle measurements in degrees or radians, it's important to ensure you're calculating using the correct unit.
For example, to find \( \sec 30^{\circ} \), first find the result of \( \cos 30^{\circ} \), and take its reciprocal.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are derived from the basic sine, cosine, and tangent functions. Each of these functions has a reciprocal:

• The reciprocal of sine is cosecant, written as \( \csc \theta = \frac{1}{\sin \theta} \).
• The reciprocal of cosine is secant, \( \sec \theta = \frac{1}{\cos \theta} \).
• The reciprocal of tangent is cotangent, expressed as \( \cot \theta = \frac{1}{\tan \theta} \).

These reciprocal functions are widely used in trigonometry to solve different equations and are critical when dealing with right triangles, circles, and various applications in calculus.

Cosine

The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse. It is one of the primary trigonometric functions.
Cosine values for common angles can be remembered quickly for use without a calculator. For instance, \( \cos 30^{\circ} \) is known to be approximately \( 0.8660 \).
Cosine, along with sine and tangent, is integral in trigonometric identities and solving triangles. Understanding these basic values makes it easier to work with reciprocal trigonometric functions like secant.

Rounding Numbers

Rounding numbers is the process of approximating a number to make it simpler to use, especially when precision is not essential, or when working with significant digits.
In mathematics, and particularly in trigonometry, it's common to round off to a certain number of decimal places, ensuring calculations remain consistent.
When rounding to four decimal places, you observe the fifth decimal place. If it is 5 or more, round up the fourth decimal, otherwise, leave it as it is. For the example \( 1.1557 \), it was rounded from an exact reciprocal that initially might have had more decimal places.