Problem 60 - Algebra 2

Question

Find the exact value of each expression. \(\cos 315^{\circ}\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Angle

The angle given is \(315^{\circ}\). Notice that \(315^{\circ}\) is greater than \(270^{\circ}\) but less than \(360^{\circ}\), so it lies in the fourth quadrant of the unit circle.

Step 2: Identify the Reference Angle

Since \(315^{\circ}\) is in the fourth quadrant, you determine the reference angle by subtracting \(315^{\circ}\) from \(360^{\circ}\): \(360^{\circ} - 315^{\circ} = 45^{\circ}\). Thus, the reference angle is \(45^{\circ}\).

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

In the fourth quadrant, the cosine function is positive because it represents the horizontal component of the unit circle.

Step 4: Use the Reference Angle to Find the Cosine

The cosine of \(45^{\circ}\), the reference angle, is \(\frac{\sqrt{2}}{2}\). Since cosine is positive in the fourth quadrant, \(\cos 315^{\circ} = \frac{\sqrt{2}}{2}\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle of radius 1, centered at the origin of the coordinate plane, where every point on the circle corresponds to a specific angle and its respective sine and cosine values. The x-coordinate of a point on the unit circle is the cosine of the angle, and the y-coordinate is the sine of the angle. This makes the unit circle a powerful tool for understanding the relationships between angles and their trigonometric functions.

The unit circle is divided into four quadrants, with each quadrant corresponding to angles in specific ranges:

• First Quadrant: 0 to 90 degrees
• Second Quadrant: 90 to 180 degrees
• Third Quadrant: 180 to 270 degrees
• Fourth Quadrant: 270 to 360 degrees

This setup helps determine the signs of trigonometric functions: cosine and sine have specific signs depending on the angle's quadrant.

Cosine Function

The cosine function, denoted as \( ext{cos}(\theta)\), represents the horizontal coordinate of a point on the unit circle. As the angle \(\theta\) moves around the unit circle, the cosine value changes between -1 and 1.

Cosine values are straightforward to find for common angles like 0°, 30°, 45°, 60°, and 90°, where memorizing can be helpful. For instance, for 45°, the cosine is \(\frac{\sqrt{2}}{2}\). A key feature of the cosine function is that it is even, meaning \( ext{cos}(-\theta) = \text{cos}(\theta)\).

Note that in the unit circle:

• First and Fourth Quadrants: The cosine is positive.
• Second and Third Quadrants: The cosine is negative.

In the case of our exercise, since 315° lies in the fourth quadrant, the cosine value will be positive.

Reference Angle

A reference angle is the smallest angle that a given angle makes with the x-axis. It is always measured between 0° and 90°. Finding the reference angle is crucial because it helps determine the trigonometric function's value for any angle by leveraging known values from the acute angles.

To find a reference angle:

• 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 given exercise, the angle 315° in the fourth quadrant has a reference angle of 45° (calculated as 360° - 315°). It's helpful because the cosine of 45° is a common trigonometric value, \(\frac{\sqrt{2}}{2}\), and applies with the correct sign based on the quadrant.

Quadrants in Trigonometry

Understanding quadrants in trigonometry is essential since they dictate the signs of the trigonometric functions. The unit circle is divided into four quadrants:

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

The quadrant an angle lies in can be quickly identified by its degree measure or radian measure. For example, the angle 315° lies in the fourth quadrant where, as mentioned, cosine is positive. Trig angles in a specific quadrant often share trigonometric properties, which simplify computations and help in predicting the outcome of function evaluations.