Problem 43 - Precalculus with Calculus Previews

Question

Find the given trigonometric function value. Do not use a calculator. $$ \cos 330^{\circ} $$

Step-by-Step Solution

Answer

\(\cos 330^{\circ} = \frac{\sqrt{3}}{2}\)

Step 1: Identify the Reference Angle

The angle \(330^{\circ}\) is in the fourth quadrant. The reference angle can be found by subtracting \(330^{\circ}\) from \(360^{\circ}\). Thus, the reference angle is \(360^{\circ} - 330^{\circ} = 30^{\circ}\).

Step 2: Determine the Cosine Value of the Reference Angle

The cosine of \(30^{\circ}\) is a known value, \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\).

Step 3: Determine the Sign of the Cosine in the Fourth Quadrant

In the fourth quadrant, the cosine function is positive. Therefore, \(\cos 330^{\circ} = \cos 30^{\circ}\).

Step 4: Combine Information to Find \(\cos 330^{\circ}\)

Since the reference angle is \(30^{\circ}\) and the cosine is positive in the fourth quadrant, \(\cos 330^{\circ} = \frac{\sqrt{3}}{2}\).

Key Concepts

Understanding Quadrants

In the world of trigonometry, the coordinate plane is divided into four sections called quadrants. Each quadrant represents a unique range of angle measures and has different sign conventions for the trigonometric functions.

• The First Quadrant covers angles between 0° and 90°. Here, all trigonometric functions are positive.
• The Second Quadrant includes angles from 90° to 180°. In this region, sine is positive while cosine and tangent are negative.
• The Third Quadrant spans 180° to 270°, where tangent is positive, but sine and cosine are negative.
• The Fourth Quadrant involves angles from 270° to 360°. In this quadrant, cosine is positive, but sine and tangent become negative.

Identifying the quadrant helps in determining the sign of trigonometric values without needing intensive calculations or a calculator.

Mastering Reference Angles

Reference angles are crucial when working with trigonometric functions beyond the first quadrant. A reference angle is the acute angle (less than 90°) that a standard position angle makes with the x-axis.

• To find the reference angle, always take the smallest angle that can bring the given angle to the x-axis.
• For angles in the Fourth Quadrant, the reference angle is found by subtracting your angle from 360°. For example, to find the reference angle of 330°, you calculate: 360° - 330° = 30°.
• Knowing this 30° reference angle, you can then use the known values of trigonometric functions for 30° to solve the problem.

The concept of reference angles allows simplified calculation, as you use familiar acute angle values to determine trigonometric function results for angles in other quadrants.

Angle Measurement Basics

Angles form the foundation of trigonometry, and understanding them is essential for solving problems involving trigonometric identities. Angles are measured either in degrees or radians, with 360° equivalent to a full circle.

• In trigonometry problems, knowing how to convert between angle units is important. For simplicity, most introductory problems use degrees.
• Angles in trigonometry can define a position on the unit circle, helping determine various trigonometric function values.
• The measurement also helps to determine whether an angle is within a particular quadrant and hence, the sign of a trigonometric ratio.

By understanding angle measurements, students can accurately work through trigonometry problems and correctly identify resulting values within specific boundaries of the circle.