Problem 87 - A Graphical Approach to Precalculus with Limits

Question

Use a reference angle to find \(\sin \theta\) and \(\cos \theta\) for the given \(\theta\). $$\theta=240^{\circ}$$

Step-by-Step Solution

Answer

\(\sin 240^{\circ} = -\frac{\sqrt{3}}{2}\) and \(\cos 240^{\circ} = -\frac{1}{2}\).

Step 1: Identify the Quadrant

The angle \(\theta = 240^{\circ}\) is in the third quadrant because it falls between \(180^{\circ}\) and \(270^{\circ}\).

Step 2: Find the Reference Angle

The reference angle \(\theta_{ref}\) is the angle between \(\theta\) and the x-axis. In the third quadrant, the reference angle is found by calculating \(\theta - 180^{\circ}\). Thus, \(\theta_{ref} = 240^{\circ} - 180^{\circ} = 60^{\circ}\).

Step 3: Determine Signs of Sine and Cosine

In the third quadrant, both sine and cosine are negative. This is important for determining the signs of \(\sin \theta\) and \(\cos \theta\) for \(\theta = 240^{\circ}\).

Step 4: Evaluate Sine and Cosine

Using the reference angle \(60^{\circ}\), we know \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\) and \(\cos 60^{\circ} = \frac{1}{2}\). Since both sine and cosine are negative in the third quadrant, we have \(\sin 240^{\circ} = -\frac{\sqrt{3}}{2}\) and \(\cos 240^{\circ} = -\frac{1}{2}\).

Key Concepts

Reference Angle

A reference angle is a helpful concept in trigonometry that allows you to simplify the calculation of trigonometric functions for angles greater than 90 degrees. It is defined as the acute angle formed with the x-axis and is always positive. To find the reference angle:

• For angles in the first quadrant, the reference angle is the angle itself.
• For angles in the second quadrant, subtract the angle from 180 degrees.
• For angles in the third quadrant, subtract 180 degrees from the angle, as shown in our example: \[\theta_{ref} = 240^{\circ} - 180^{\circ} = 60^{\circ}\].
• For angles in the fourth quadrant, subtract the angle from 360 degrees.

An understanding of reference angles allows you to use basic trig function values even for more complex angles found outside the first quadrant.

Quadrants in Trigonometry

The coordinate system is divided into four quadrants, each influencing the sign of trigonometric functions such as sine and cosine. Here's how they work:

• First Quadrant (0° to 90°): Both sine and cosine are positive.
• Second Quadrant (90° to 180°): Sine is positive, cosine is negative.
• Third Quadrant (180° to 270°): Both sine and cosine are negative, as seen with \(\theta = 240^{\circ}\).
• Fourth Quadrant (270° to 360°): Sine is negative, cosine is positive.

When working with angles, it's crucial to identify the correct quadrant so you can determine the correct signs of trigonometric functions. It's simple once you remember these general rules for each quadrant.

Sine and Cosine Functions

The sine and cosine functions are fundamental in trigonometry. They relate the angles of a triangle to the lengths of its sides in the unit circle.For any angle \(\theta\), the values of sine and cosine can be determined using reference angles and the quadrants they reside in. For example:

• The sine of 60° is \(\frac{\sqrt{3}}{2}\), and cosine is \(\frac{1}{2}\).
• For \(\theta = 240^{\circ}\) in the third quadrant, these values are negative, so \(\sin 240^{\circ} = -\frac{\sqrt{3}}{2}\) and \(\cos 240^{\circ} = -\frac{1}{2}\).

Understanding that trigonometric functions repeat their values in a cyclical pattern, depending on the angle and quadrants, helps simplify complex trigonometric calculations. This cyclical pattern means you only need to know the sine and cosine in the first quadrant to determine them elsewhere using reference angles and quadrant rules.