Problem 90 - A Graphical Approach to Precalculus with Limits

Question

Use a reference angle to find \(\sin \theta\) and \(\cos \theta\) for the given \(\theta\). $$\theta=-45^{\circ}$$

Step-by-Step Solution

Answer

\(\sin(-45^{\circ}) = -\frac{\sqrt{2}}{2}\), \(\cos(-45^{\circ}) = \frac{\sqrt{2}}{2}\).

Step 1: Determine the reference angle

The reference angle is the positive angle that is formed by the terminal side of the given angle and the x-axis. Since \(-45^{\circ}\) is already within 0 to \(-90^{\circ}\), the reference angle is \(|-45^{\circ}| = 45^{\circ}\).

Step 2: Use the unit circle or known values

In a unit circle, the sine and cosine of \(45^{\circ}\) are \(\sin 45^{\circ} = \frac{\sqrt{2}}{2}\) and \(\cos 45^{\circ} = \frac{\sqrt{2}}{2}\).

Step 3: Adjust signs according to the angle's quadrant

The angle \(-45^{\circ}\) is located in the fourth quadrant, where sine is negative and cosine is positive. Therefore, \(\sin(-45^{\circ}) = -\frac{\sqrt{2}}{2}\) and \(\cos(-45^{\circ}) = \frac{\sqrt{2}}{2}\).

Key Concepts

Reference Angles

A reference angle is a fundamental concept in trigonometry that helps us determine the trigonometric functions of any angle. It is the acute angle formed by the terminal side of the given angle and the x-axis. No matter where the original angle lies, the reference angle will always be between 0° and 90°, making calculations easier. For example, if we are given an angle like \(-45^{\circ}\), its reference angle is \(|-45^{\circ}| = 45^{\circ}\). This reference angle tells us the "distance" from the x-axis in terms of angle measurement.
By understanding reference angles, you can work with angles located in different quadrants and apply known sine and cosine values from the first quadrant. It simplifies otherwise complex calculations, especially when memorizing only the basic trigonometric values.

Unit Circle

The unit circle is a pivotal tool in trigonometry, providing a visual way to understand angles and their corresponding sine and cosine values. The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It encompasses an infinite number of angles, all of which have coordinates representing \(x = \cos \theta\) and \(y = \sin \theta\).
For instance, on the unit circle, the angle \(45^{\circ}\) corresponds to the coordinates \((\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\). These values show that both the sine and cosine of \(45^{\circ}\) are \(\frac{\sqrt{2}}{2}\). The unit circle is incredibly useful for quickly determining the sine and cosine of an angle since the x-coordinate provides cosine values, and the y-coordinate gives sine values.
Understanding the unit circle means knowing that angles can be measured not only in degrees but also in radians, offering an additional dimension to trigonometric functions.

Quadrants

The coordinate plane is split into four quadrants, each having its own set of rules for the signs of trigonometric functions. Knowing which quadrant an angle lies in helps determine the sign of its trigonometric values.

• First Quadrant: Both sine and cosine are positive.
• Second Quadrant: Sine is positive, cosine is negative.
• Third Quadrant: Both sine and cosine are negative.
• Fourth Quadrant: Sine is negative, cosine is positive.

When an angle like \(-45^{\circ}\) appears, it falls into the fourth quadrant. It means sine will be negative and cosine will be positive, reflecting the character of that quadrant.
Recognizing which quadrant an angle is in allows you to adjust the standard trigonometric values taken from the unit circle. By flipping the signs accordingly, you can find the correct trigonometric functions for any angle.