Problem 64 - Precalculus

Question

Evaluate the following expressions exactly: $$\sec \left(-330^{\circ}\right)$$

Step-by-Step Solution

Answer

\(\sec(-330°) = \frac{2\sqrt{3}}{3}\).

Step 1: Simplify the Angle

First, find an equivalent angle by adding full circle rotations (360°) to the given angle such that the equivalent angle is positive and less than 360°. The angle given is -330°, so we add 360° to find the equivalent angle:\(-330° + 360° = 30°\).

Step 2: Determine the Related Acute Angle

Since 30° is already an acute angle, the related acute angle is 30°.

Step 3: Identify the Trigonometric Ratio

The secant function is the reciprocal of the cosine function. Therefore, \( \sec(30°) = \frac{1}{\cos(30°)} \).

Step 4: Use Known Values

The cosine of 30° is a well-known value: \( \cos(30°) = \frac{\sqrt{3}}{2} \).

Step 5: Calculate the Secant

Using the known value of cosine:\( \sec(30°) = \frac{1}{\cos(30°)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \).

Step 6: Rationalize the Denominator

Rationalize the fraction so there is no square root in the denominator:\( \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \).

Key Concepts

Secant Function

The secant function, commonly denoted as \( \sec \theta \), is one of the six fundamental trigonometric functions. It is the reciprocal of the cosine function, which means it can be expressed as:

• \( \sec \theta = \frac{1}{\cos \theta} \)

Whenever you're working with the secant function, it’s important to remember the relationship between secant and cosine, as it allows you to determine the secant value using known cosine values.
This can be particularly useful when working with angles commonly found in triangles and unit circle trigonometry like 30°, 45°, and 60°.

For example, knowing that \( \cos(30°) = \frac{\sqrt{3}}{2} \), we can find \( \sec(30°) \) by directly calculating its reciprocal:

• \( \sec(30°) = \frac{1}{\cos(30°)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \)

Understanding the secant function is essential for simplifying complex trigonometric expressions and solving trigonometric equations.

Angle Simplification

Angle simplification involves rewriting a given angle to its equivalent value within a specified range, usually between 0° and 360° for degree measures, to make calculations easier and more intuitive.
This process is crucial when dealing with negative angles or angles that exceed 360°, because they occur as full circle rotations, and the trigonometric functions repeat their behaviour after every full rotation of 360°.

For instance, in this exercise, we have a negative angle of -330°;
To simplify this, we add 360° to the angle to find its equivalent within a normal circle range:

• \( -330° + 360° = 30° \)

The simplified angle is 30°, which is easier to work with as it fits within our familiar range of the unit circle, enabling us to use known trigonometric values to solve expressions.
These steps greatly simplify the understanding and solving of trigonometric problems by making angles more manageable.

Rationalization

Rationalization is a mathematical technique used to eliminate square roots or irrational numbers from the denominator of a fraction.
This process simplifies expressions, making them easier to interpret and work with in subsequent mathematical operations.

In trigonometry, when calculating trigonometric functions like secant, you might end up with a square root in the denominator, as seen in the calculation of \( \sec(30°) = \frac{2}{\sqrt{3}} \).

To rationalize:

• Multiply both the numerator and the denominator by the square root present in the denominator:
• \( \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \)

This rationalized form \( \frac{2\sqrt{3}}{3} \) is considered more spatially efficient in terms of computation and presentation.
Rationalization hence plays a significant role in optimizing and refining mathematical expressions, paving the way for smoother and clearer calculations.