Problem 17 - Algebra 2

Question

Find the exact value of each expression. \(\cos 225^{\circ}\)

Step-by-Step Solution

Answer

\( \cos 225^\circ = -\frac{\sqrt{2}}{2} \)

Step 1: Identify Reference Angle and Quadrant

First, note that the angle 225° is in the third quadrant. To find the reference angle, subtract from the nearest x-axis angle, which is 180°: 225° - 180° = 45°. Thus, the reference angle is 45°.

Step 2: Determine Cosine Value in Third Quadrant

In the third quadrant, the cosine value is negative because the x-coordinates are negative. Since the reference angle is 45°, first consider the cosine of 45°, which is \( \cos 45^\circ = \frac{\sqrt{2}}{2} \). Thus, \( \cos 225^\circ = -\cos 45^\circ = -\frac{\sqrt{2}}{2} \).

Key Concepts

Reference Angle

The reference angle is instrumental when dealing with trigonometric functions, as it helps simplify the process of finding angles that are not primary angles like 0°, 30°, 45°, 60°, and 90°. The reference angle is the smallest angle that the terminal side of an angle makes with the x-axis.
To find the reference angle for any angle, you need to determine the angle's location within one of the four quadrants of the Cartesian plane. When an angle, like 225°, falls beyond the first quadrant, finding the reference angle involves:

• Identifying the angle's quadrant: 225° is in the third quadrant because it is between 180° and 270°.
• Calculating its reference angle: Subtract the nearest x-axis angle. For the third quadrant, you subtract 180°, so 225° - 180° = 45°.

With the reference angle of 45°, you can easily determine trigonometric values using known values of primary angles.

Trigonometric Functions

Trigonometric functions relate the angles of a triangle to the ratios of its sides. The primary functions are sine (\(\sin\theta\)), cosine (\(\cos\theta\)), and tangent (\(\tan\theta\)). Each function represents a specific ratio:

• Cosine, or \(\cos\theta\), is the ratio of the adjacent side to the hypotenuse in a right triangle.
• Sine, or \(\sin\theta\), is the ratio of the opposite side to the hypotenuse.
• Tangent, or \(\tan\theta\), is the ratio of the opposite side to the adjacent side.

The value of these functions changes depending on the quadrant in which the angle resides. This property of trigonometric functions is crucial for determining the signs of function values; in the third quadrant, the cosine function is intrinsically negative. Therefore, knowing your reference angle, like 45°, assists in using these functions effectively to find exact angle values.

Third Quadrant

The third quadrant of the unit circle is where both x and y coordinates are negative. Understanding this quadrant's properties is crucial, as they impact the trigonometric function values:

• In the third quadrant, angles range between 180° and 270°.
• The cosine of any angle here is negative, while sine is also negative because both the x- and y- coordinates on the unit circle are negative.
• Tangent, however, is positive in this quadrant since it is the ratio of sine to cosine, i.e., \(\tan\theta = \frac{-y}{-x} = \frac{y}{x}\).

For 225°, which lies in the third quadrant, you apply these sign rules. Since the cosine of 45°, the reference angle, is \(\frac{\sqrt{2}}{2}\), the cosine of 225°, incorporating the negative sign from the third quadrant, results in \(-\frac{\sqrt{2}}{2}\). Understanding the quadrant's influence is vital to finding precise trigonometric values and avoiding common mistakes.