Problem 20 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function. $$ \sec 120^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \( \sec 120^\circ \) is \(-2\).

Step 1: Convert Degree to Radian

The given angle is in degrees. We must first verify if it is eligible for standard trigonometric identities. For angles in degrees like 120°, this is directly usable in the standard trigonometric functions. Thus 120° is to be evaluated directly without conversion to radians in this case.

Step 2: Use Trigonometric Identity

Recall the trigonometric identity that relates \( \sec \theta \) to the cosine function: \(\sec \theta = \frac{1}{\cos \theta}. \)We need to find \( \cos 120^\circ \) first.

Step 3: Find Cosine of 120°

The angle \(120^\circ\) is in the second quadrant, where cosine is negative. The reference angle for \(120^\circ\) is \(180^\circ - 120^\circ = 60^\circ\).Thus, \(\cos 120^\circ = -\cos 60^\circ = -\frac{1}{2}.\)

Step 4: Calculate Secant

Using the identity \( \sec \theta = \frac{1}{\cos \theta}, \) substitute the value of \(\cos 120^\circ\): \[\sec 120^\circ = \frac{1}{-\frac{1}{2}} = -2.\]]]},{

Step 5: State the Exact Value

Therefore, the exact value of \( \sec 120^\circ \) is \(-2\).

Key Concepts

Secant Function

The secant function is one of the six main trigonometric functions. It is the reciprocal of the cosine function. More precisely, if you have an angle \( \theta \), the secant of \( \theta \) is defined as follows:

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

This formula shows that wherever the cosine is zero, the secant function becomes undefined since division by zero is impossible. The secant function is useful for identifying distances in trigonometric contexts and finds application across various fields in mathematics and physics.
When solving for \( \sec 120^{\circ} \), understanding its relation to cosine is key, as demonstrated in the problem solution. This connection allows for a straightforward calculation by first determining \( \cos 120^{\circ} \) and then using the reciprocal relationship.

Cosine Function

The cosine function is fundamental in trigonometry, often represented as \( \cos \theta \). It describes the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Importantly, cosine values are affected by the angle's position on the unit circle.

• In the first quadrant, cosine values are positive.
• In the second quadrant, cosine values become negative, as shown with \( \cos 120^{\circ} = -\frac{1}{2} \).
• In the third and fourth quadrants, cosine values also vary between negative and positive, respectively.

When dealing with angles greater than \(90^{\circ}\), like \(120^{\circ}\), it’s crucial to recognize which quadrant the angle lies in. For \(120^{\circ}\), it’s in the second quadrant, where cosine is negative. This specific feature enables correct evaluation and simplification, ensuring that computations for functions like secant are precise.

Trigonometric Identities

Trigonometric identities are equations that are true for all angular values where both sides of the equation are defined. These identities form the backbone of simplifying trigonometric expressions and solving trigonometric equations.

• The reciprocal identities, such as \( \sec \theta = \frac{1}{\cos \theta} \), help in evaluating secants based on cosine values.
• Other identities like the Pythagorean identities, addition formulas, and double-angle formulas, allow for various manipulations and calculations.

Each identity has its importance and application based on the problem you are working on. In the case of \( \sec 120^{\circ} \), the reciprocal identity directly facilitates finding its value by relating it to the corresponding cosine value. Recognizing and applying these identities is crucial for advanced trigonometric problems, helping simplify and solve them efficiently.