Problem 88 - A Graphical Approach to Precalculus with Limits

Question

Use a reference angle to find \(\sin \theta\) and \(\cos \theta\) for the given \(\theta\). $$\theta=315^{\circ}$$

Step-by-Step Solution

Answer

\( \sin 315^{\circ} = -\frac{\sqrt{2}}{2} \), \( \cos 315^{\circ} = \frac{\sqrt{2}}{2} \).

Step 1: Understand the Angle

To find the reference angle, we first need to identify the location of the angle \( \theta = 315^{\circ} \) on the unit circle. This angle is in the fourth quadrant because it is greater than \( 270^{\circ} \) and less than \( 360^{\circ} \).

Step 2: Find the Reference Angle

The reference angle \( \theta_{ref} \) for an angle in the fourth quadrant is given by \( 360^{\circ} - \theta \). Substituting the values, we get \( 360^{\circ} - 315^{\circ} = 45^{\circ} \). Thus, the reference angle is \( 45^{\circ} \).

Step 3: Use the Reference Angle to Determine the Sine

The sine of an angle in the fourth quadrant has the same magnitude as the sine of its reference angle, but it is negative. Therefore, \( \sin \theta = -\sin 45^{\circ} = -\frac{\sqrt{2}}{2} \).

Step 4: Use the Reference Angle to Determine the Cosine

The cosine of an angle in the fourth quadrant has the same magnitude as the cosine of its reference angle because cosine is positive in this quadrant. Thus, \( \cos \theta = \cos 45^{\circ} = \frac{\sqrt{2}}{2} \).

Key Concepts

Reference Angle

When dealing with trigonometric functions, the reference angle is an essential concept. It helps simplify the calculation of \(\sin \theta\) and \(\cos \theta\) for angles that are not positioned in the first quadrant. The reference angle is the acute angle formed by the terminal side of the given angle and the horizontal axis. Its value is always between 0° and 90°.

For an angle residing in different quadrants, here's how the reference angle is calculated:

• First Quadrant: The reference angle is the angle itself.
• Second Quadrant: Subtract the angle from 180°.
• Third Quadrant: Subtract 180° from the angle.
• Fourth Quadrant: Subtract the angle from 360°.

In the case of \(\theta = 315^{\circ}\), the angle is in the fourth quadrant, so the reference angle is\[360^{\circ} - 315^{\circ} = 45^{\circ}.\]This reference angle, \(45^{\circ}\), is used to find the values of sine and cosine.

Unit Circle

The unit circle is a critical tool in trigonometry and is used to define the sine, cosine, and tangent functions. It's a circle with its center at the origin of the coordinate plane and a radius of 1.

Angles on the unit circle are measured in radians or degrees starting from the positive x-axis. Here are some key points about the unit circle:

• Quadrants: The circle is divided into four quadrants, each corresponding to different signs for trigonometric functions.
• Coordinates: Every point on the unit circle corresponds to \((\cos \theta, \sin \theta)\).
• Special Angles: Angles like 45°, 90°, 135°, and so on, have known sine and cosine values, which simplify calculations.

Knowing that 315° is in the fourth quadrant, we can use the unit circle to determine that:

• The cosine of 315° is positive because cosine values are positive in the fourth quadrant.
• The sine of 315° is negative since sine values are negative in the fourth quadrant.

These insights come directly from understanding the unit circle's application.

Sine and Cosine Values

To determine the sine and cosine values for any angle, especially when the angle is not in the first quadrant, you use the reference angle as the stepping stone.

For 315°, its reference angle is 45°, and hence:

• The sine of 45° is \(\frac{\sqrt{2}}{2}\). However, in the fourth quadrant, sine is negative, leading to \(\sin 315^{\circ} = -\frac{\sqrt{2}}{2}\).
• Similarly, the cosine of 45° is \(\frac{\sqrt{2}}{2}\), and since cosine is positive in the fourth quadrant, \(\cos 315^{\circ} = \frac{\sqrt{2}}{2}\).

Understanding these values is made easier by revisiting the unit circle. Remember that the signs of sine and cosine depend on the specific quadrant. Here, memorizing the behavior of these functions and the symmetry in the unit circle will be incredibly helpful.