Problem 20 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

\(\sec 120^{\circ} = -2\).

Step 1: Understanding the Angle

The angle given is \(120^{\circ}\). This angle is in the second quadrant of the unit circle, where the cosine function is negative.

Step 2: Find the Reference Angle

To find the reference angle for \(120^{\circ}\), subtract \(120^{\circ}\) from \(180^{\circ}\): \(180^{\circ} - 120^{\circ} = 60^{\circ}\). Therefore, the reference angle is \(60^{\circ}\).

Step 3: Calculate Cosine of 120°

From the unit circle, the cosine of \(60^{\circ}\) is \(\frac{1}{2}\). Since \(120^{\circ}\) is in the second quadrant where cosine is negative, \(\cos 120^{\circ} = -\frac{1}{2}\).

Step 4: Determine Secant from Cosine

The secant function is the reciprocal of the cosine function. So, \(\sec 120^{\circ} = \frac{1}{\cos 120^{\circ}}\).

Step 5: Calculate the Exact Value

Calculate the exact value of \(\sec 120^{\circ}\) using the reciprocal of cosine: \[\sec 120^{\circ} = \frac{1}{-\frac{1}{2}} = -2\].

Key Concepts

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. It is a crucial tool in trigonometry as it provides a way to define trigonometric functions for all angles.

• Each point on the unit circle corresponds to an angle \( \theta \) measured from the positive x-axis.
• The coordinates of each point are \( (\cos \theta, \sin \theta) \).

For example, an angle of \( 120^{\circ} \) on the unit circle has coordinates that help us find the cosine and sine values.Understanding the unit circle can help visualize how trigonometric functions behave as angles progress through different quadrants.

Second Quadrant

The unit circle is divided into four quadrants, with each quadrant representing a range of angles.

• The second quadrant lies between \( 90^{\circ} \) and \( 180^{\circ} \).
• In this quadrant, sine values are positive, while cosine values are negative.

Knowing which function is positive or negative in the various quadrants is essential when solving trigonometric problems. When dealing with an angle like \( 120^{\circ} \), it is important to note that it resides in the second quadrant, thus influencing the sign of cosine, as indicated by the solution's finding \( \cos 120^{\circ} = -\frac{1}{2} \).

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are those that involve flipping the numerator and the denominator of the original trigonometric ratios. The primary functions include:

• Cosecant (\( \csc \theta \)), which is the reciprocal of sine: \( \csc \theta = \frac{1}{\sin \theta} \).
• Secant (\( \sec \theta \)), the reciprocal of cosine: \( \sec \theta = \frac{1}{\cos \theta} \).
• Cotangent (\( \cot \theta \)), which serves as the reciprocal of tangent: \( \cot \theta = \frac{1}{\tan \theta} \).

These functions are particularly useful when determining ratios that are undefined for certain angles due to division by zero.For example, in the exercise, \( \sec 120^{\circ} = \frac{1}{\cos 120^{\circ}} = -2 \). Understanding reciprocal functions enables handling a wide range of trigonometric problems efficiently.