Problem 95 - Precalculus

Question

Evaluate the expression \(\sec 120^{\circ}\) exactly. Solution: \(120^{\circ}\) lies in quadrant 11 The reference angle is \(30^{\circ}\) \(\begin{array}{l}\text { Find the cosine of the } \\ \text { reference angle. }\end{array} \quad \cos 30^{\circ}=\frac{\sqrt{3}}{2}\) \(\begin{array}{l}\text { Cosine is negative in } \\ \text { quadrant II. }\end{array} \quad \cos 120^{\circ}=-\frac{\sqrt{3}}{2}\) \(\begin{array}{l}\text { Secant is the reciprocal } \\ \text { of cosine. }\end{array} \quad \sec 120^{\circ}=-\frac{2}{\sqrt{3}}=-\frac{2 \sqrt{3}}{3}\) This is incorrect. What mistake was made? (GRAPH CANNOT COPY)

Step-by-Step Solution

Answer

The mistake was using \( \cos 30^{\circ} \) instead of \( \cos 60^{\circ} \); use the correct reference angle to find \( \sec 120^{\circ} = -2 \).

Step 1: Identify Quadrant and Reference Angle

To evaluate \( \sec 120^{\circ} \), first identify that \( 120^{\circ} \) is in the second quadrant. The reference angle for \( 120^{\circ} \) is \( 180^{\circ} - 120^{\circ} = 60^{\circ} \). This is because the reference angle is measured from the nearest horizontal axis.

Step 2: Find Cosine of the Reference Angle

Next, find the cosine of the reference angle \( 60^{\circ} \). We know that \( \cos 60^{\circ} = \frac{1}{2} \).

Step 3: Determine Sign Based on Quadrant

Since cosine is negative in the second quadrant, we have to make \( \cos 120^{\circ} = -\cos 60^{\circ} = -\frac{1}{2} \).

Step 4: Calculate Secant

Secant is the reciprocal of cosine. Thus, \( \sec 120^{\circ} = \frac{1}{\cos 120^{\circ}} = \frac{1}{-\frac{1}{2}} = -2 \).

Key Concepts

Secant Function

The Secant Function, denoted as \( \sec \theta \), is one of the six fundamental trigonometric functions. It is closely related to the cosine function. In fact, the secant is simply the reciprocal of the cosine function.

• Mathematically, \( \sec \theta = \frac{1}{\cos \theta} \).
• This means that if \( \cos \theta \) is known, the secant can be easily calculated by taking its reciprocal.

Taking the reciprocal of a value involves swapping the numerator and the denominator in a fraction. Therefore, when evaluating expressions like \( \sec 120^{\circ} \), we primarily focus on determining the value of \( \cos 120^{\circ} \) first and then find its reciprocal to get the secant value.Additionally, the secant function is undefined where the cosine value is zero, as division by zero is not possible.

Quadrants in Trigonometry

Trigonometry divides the cartesian plane into four distinct quadrants. These quadrants help us understand the behavior of trigonometric functions based on the angle's location.

• Quadrant I contains angles between \( 0^{\circ} \) and \( 90^{\circ} \). In this quadrant, all trigonometric functions are positive.
• Quadrant II, from \( 90^{\circ} \) to \( 180^{\circ} \), has sine as positive and cosine and secant as negative.
• Quadrant III spans \( 180^{\circ} \) to \( 270^{\circ} \), where tangent and cotangent are positive.
• Quadrant IV covers angles \( 270^{\circ} \) to \( 360^{\circ} \), making cosine positive again.

When solving trigonometric problems, identifying the correct quadrant is essential, as it determines the sign (positive or negative) of the trigonometric function being evaluated. For example, since \( 120^{\circ} \) lies in Quadrant II, the cosine and therefore the secant will be negative.

Reference Angle

A Reference Angle is the acute angle that a given angle makes with the horizontal axis. It is always between \( 0^{\circ} \) and \( 90^{\circ} \), acting as a handy tool to simplify trigonometric calculations.To find the reference angle:

• For angles in Quadrant I, the reference angle is the angle itself.
• In Quadrant II, compute \( 180^{\circ} - \theta \).
• For Quadrant III, find \( \theta - 180^{\circ} \).
• And in Quadrant IV, calculate \( 360^{\circ} - \theta \).

For \( 120^{\circ} \), which is in Quadrant II, the reference angle is \( 180^{\circ} - 120^{\circ} = 60^{\circ} \). Once we have the reference angle, we can easily find standard trigonometric values such as sine and cosine, aiding in solving the problem more efficiently.

Cosine Function

The Cosine Function, denoted as \( \cos \theta \), is another fundamental trigonometric function that relates the angle in a right triangle to the ratio of the adjacent side to the hypotenuse.

• For special angles like \( 60^{\circ} \), \( \cos 60^{\circ} = \frac{1}{2} \).
• This is a universally known value and is derived from equilateral triangles or using the unit circle.

Knowing these values is particularly useful when determining other related trigonometric functions, such as secant. In solving \( \sec 120^{\circ} \), we found \( \cos 120^{\circ} \) first, which was \( -\frac{1}{2} \) given the negative sign from the second quadrant. Understanding the properties of the cosine function and recognizing its transformation in various quadrants is fundamental in trigonometry.