Problem 16 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function. $$ \sec 300^{\circ} $$

Step-by-Step Solution

Answer

\( \sec 300^{\circ} = 2 \)

Step 1: Identify the Reference Angle

The angle given is greater than 360°, so we first find an equivalent angle by subtracting 360° from it. However, since 300° is less than 360°, there is no need to adjust it. The reference angle for 300° can be found by subtracting 300° from 360°, which gives us 60°.

Step 2: Determine the Quadrant

The angle 300° is in the fourth quadrant. In this quadrant, the cosine function is positive, which means that the secant function, as the reciprocal of cosine, is also positive.

Step 3: Calculate the Secant

The secant function is the reciprocal of the cosine function. We first find the cosine of the reference angle 60°, which is \( \cos 60^{\circ} = \frac{1}{2} \). The secant of 300° is then the reciprocal of this value: \( \sec 300^{\circ} = \frac{1}{\cos 300^{\circ}} = \frac{1}{\frac{1}{2}} = 2 \).

Step 4: Verify the Sign

In the fourth quadrant, cosine is positive, so secant will also be positive. Therefore, \( \sec 300^{\circ} = 2 \) is correct as it is calculated with the positive cosine value.

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is one of the six fundamental trigonometric functions. It is related to the cosine function and is defined as the reciprocal of the cosine function. In simple terms, if you have a cosine of an angle, the secant is 1 divided by that cosine value. Mathematically, it can be expressed as: \[ \sec \theta = \frac{1}{\cos \theta} \] The secant function is particularly useful when you need to calculate ratios that involve the hypotenuse over the adjacent side in a right triangle, but aren't directly known via cosine.

• The secant function is undefined whenever the cosine of the angle is zero because division by zero is mathematically undefined.
• Angles where cosine equals zero include 90°, 270°, and any odd multiples of these angles.
• It is important to pay attention to the sign of the cosine value, as it will directly affect the sign of the secant.

Understanding the secant function helps in solving trigonometric problems involving non-standard angles, as well as in graphing and analysis of functions.

Reference Angle

A reference angle is crucial for simplifying the evaluation of trigonometric functions for certain angles. The reference angle is the acute angle formed by the terminal side of the given angle and the horizontal axis. It is always positive and between 0° and 90°. For an angle \( \theta \), the reference angle \( \theta_r \) can be calculated as:

• If \( \theta \) lies in the first quadrant, then \( \theta_r = \theta \).
• If \( \theta \) lies in the second quadrant, then \( \theta_r = 180° - \theta \).
• If \( \theta \) lies in the third quadrant, then \( \theta_r = \theta - 180° \).
• If \( \theta \) lies in the fourth quadrant, then \( \theta_r = 360° - \theta \).

Using the reference angle, we can easily determine the trigonometric function values, as the function will have the same magnitude as that of its reference angle, modulo sign. This conceptual strategy greatly aids in evaluating angles greater than 90° and less than 360°, as it leverages symmetrical properties of the unit circle.

Quadrants in Trigonometry

The coordinate plane is divided into four sections known as quadrants in trigonometry, each corresponding to a particular range of angle measures. These quadrants help determine the sign of trigonometric functions. Understanding these divisions is crucial when solving trigonometric equations or evaluating trigonometric functions at certain angles.

• The First Quadrant spans 0° to 90°, where all trigonometric functions (sine, cosine, tangent, etc.) are positive.
• The Second Quadrant covers 90° to 180°. Here, sine is positive while cosine and tangent are negative.
• The Third Quadrant lies between 180° and 270°, where tangent is positive and both sine and cosine are negative.
• The Fourth Quadrant ranges from 270° to 360°. In this quadrant, cosine is positive, while sine and tangent are negative.

Knowing which quadrant an angle is in allows us to determine the sign of any trigonometric function at that angle without direct calculation. This is why, for instance, the secant of 300° is positive; because 300° lies in the fourth quadrant, where cosine (and thus secant) remains positive.