Question

Find the exact value of each expression. Do not use a calculator. $$ \sec 240^{\circ} $$

Short answer

The exact value of \( \text{sec}(240^{\text{°}})\) is -2.

Step-by-step solution

- Understand the Secant Function

Recall that the secant function is defined as the reciprocal of the cosine function. That is, \(\text{sec}(\theta) = \frac{1}{\text{cos}(\theta)}\). Hence, finding \( \text{sec}(240^{\text{°}})\) requires first finding \( \text{cos}(240^{\text{°}})\).

- Locate the Angle on the Unit Circle

The angle 240° is located in the third quadrant of the unit circle. In the third quadrant, the cosine value is negative.

- Reference Angle

Determine the reference angle for 240°. The reference angle is the positive acute angle measured from the x-axis to 240°. This is calculated by subtracting 180° from 240°: \(240^{\text{°}} - 180^{\text{°}} = 60^{\text{°}}\).

- Find the Cosine of the Reference Angle

The reference angle is 60°, whose cosine value is \( \text{cos}(60^{\text{°}}) = \frac{1}{2} \). Since 240° is in the third quadrant, the cosine value for 240° will be negative: \( \text{cos}(240^{\text{°}}) = -\frac{1}{2} \).

- Calculate the Secant

Use the definition of secant to find \( \text{sec}(240^{\text{°}})\): \( \text{sec}(240^{\text{°}}) = \frac{1}{\text{cos}(240^{\text{°}})} = \frac{1}{-\frac{1}{2}} = -2 \).

Key concepts

Unit Circle

The unit circle is a critical concept in trigonometry. It is a circle with a radius of one unit, centered at the origin of a coordinate plane. This simple circle allows us to define all trigonometric functions for angles from 0° to 360°. Each point on the unit circle corresponds to an angle, and the coordinates of that point are \( (\cos(\theta), \sin(\theta)) \). The unit circle helps visualize how the sine and cosine values are derived for any given angle. For example, at 0° or 360°, the coordinates are (1, 0), meaning \( \cos(0) = 1 \) and \( \sin(0) = 0 \). Similarly, at 90°, the coordinates are (0, 1), illustrating that \( \cos(90^{\circ}) = 0 \) and \( \sin(90^{\circ}) = 1 \). Understanding the unit circle makes it easier to determine the values of trigonometric functions at different angles.

Reference Angle

A reference angle is a positive acute angle formed by the terminal side of the given angle and the x-axis. It is essential because it allows us to find the sine, cosine, and tangent of any angle based on the acute (less than 90°) version of that angle. To find the reference angle:

• If the angle is in the first quadrant (0° to 90°), it is its own reference angle.
• If the angle is in the second quadrant (90° to 180°), subtract the angle from 180°.
• If the angle is in the third quadrant (180° to 270°), subtract 180° from the angle.
• If the angle is in the fourth quadrant (270° to 360°), subtract the angle from 360°.

So for an angle of 240°, the reference angle is 240° - 180° = 60°.

Cosine Function

The cosine function, written as \( \cos(\theta) \), is a fundamental trigonometric function. It represents the x-coordinate of a point on the unit circle for a given angle \( \theta \). The cosine of an angle changes its sign based on the quadrant in which the angle lies:

• First quadrant: Cosine is positive.
• Second quadrant: Cosine is negative.
• Third quadrant: Cosine is negative.
• Fourth quadrant: Cosine is positive.

For instance, in the third quadrant, where our angle 240° is located, the cosine function is negative. Given the reference angle is 60°, and \( \cos(60^{\circ}) = \frac{1}{2} \), the cosine value for 240° is \( -\frac{1}{2} \).

Trigonometric Functions

Trigonometric functions include sine, cosine, tangent, cosecant, secant, and cotangent. These functions relate the angles of a triangle to the lengths of its sides. Trigonometric functions can be defined using the unit circle:

• Sine (\text{sin}): y-coordinate of the point on the unit circle.
• Cosine (\text{cos}): x-coordinate of the point on the unit circle.
• Tangent (\text{tan}): The ratio of y-coordinate to x-coordinate, or \( \text{tan}(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).
• Cosecant (\text{csc}): Reciprocal of sine, \( \text{csc}(\theta) = \frac{1}{\sin(\theta)} \).
• Secant (\text{sec}): Reciprocal of cosine, \( \text{sec}(\theta) = \frac{1}{\cos(\theta)} \).
• Cotangent (\text{cot}): Reciprocal of tangent, \( \text{cot}(\theta) = \frac{1}{\tan(\theta)} \).

For example, to find \( \sec(240^{\circ}) \), you need the cosine value for 240°. Since \( \cos(240^{\circ}) = -\frac{1}{2} \), just take the reciprocal: \( \text{sec}(240^{\circ}) = \frac{1}{-\frac{1}{2}} = -2 \). This shows how interrelated trigonometric functions are and why understanding them is essential.