Problem 18 - Algebra 2

Question

Find the exact value of each function. \(\cos \left(-225^{\circ}\right)\)

Step-by-Step Solution

Answer

\(-\frac{\sqrt{2}}{2}\)

Step 1: Understanding Negative Angles

Negative angles are measured clockwise from the positive x-axis. Therefore, \(-225^{ ext{°}}\) means moving 225 degrees clockwise.

Step 2: Find the Equivalent Positive Angle

Since a full circle is equal to \(360^{\circ}\), we can find a positive equivalent angle by adding 360 to \(-225^{\circ}\): \(-225^{\circ} + 360^{\circ} = 135^{\circ}\).

Step 3: Determine the Reference Angle

The positive angle \(135^{\circ}\) is located in the second quadrant. For angles in the second quadrant, the reference angle is calculated by subtracting the angle from \(180^{\circ}\):\(180^{\circ} - 135^{\circ} = 45^{\circ}\).

Step 4: Calculate the Cosine of the Reference Angle

The cosine of the reference angle \(45^{\circ}\) is \(\cos(45^{\circ}) = \frac{\sqrt{2}}{2}\). However, since \(135^{\circ}\) is in the second quadrant, cosine is negative.

Step 5: Find the Exact Value

Using the information from the reference angle and quadrant, we establish that:\(\cos(135^{\circ}) = -\frac{\sqrt{2}}{2}\). Thus, \(\cos(-225^{\circ}) = -\frac{\sqrt{2}}{2}\).

Key Concepts

Negative Angles

A negative angle is simply an angle measured in the clockwise direction from the positive x-axis. In contrast, positive angles are measured counterclockwise. For example, when you see an angle like \(-225^{\circ}\), it indicates a rotation of 225 degrees in the clockwise direction.

It's crucial to understand that negative angles are not "less than zero" in the traditional sense but are an alternate method of angle measurement. By converting negative angles to positive angles, you can often make trigonometric calculations easier. This conversion involves adding or subtracting full circle measures (360 degrees) to find an equivalent positive angle. This makes it suitable for standard trigonometric evaluations. In the context of the original problem, this conversion translates \(-225^{\circ}\) to \(135^{\circ}\).

Quadrants

The coordinate plane is divided into four quadrants, with each quadrant representing a different set of angle ranges based on the positive x-axis. Understanding which quadrant an angle lies in is important because it affects the sign (+ or -) of various trigonometric functions. Here's a brief guide:

• First Quadrant (0° to 90°): All trigonometric functions are positive.
• Second Quadrant (90° to 180°): Sine is positive, but cosine and tangent are negative.
• Third Quadrant (180° to 270°): Tangent is positive; sine and cosine are negative.
• Fourth Quadrant (270° to 360°): Cosine is positive, whereas sine and tangent are negative.

In this exercise, the equivalent positive angle is \(135^{\circ}\). This angle falls in the second quadrant, where only sine is positive, thus making the cosine value negative.

Reference Angle

A reference angle is a helpful tool in trigonometry. It represents the smallest angle between the terminal side of a given angle and the x-axis, always between 0° and 90°. Calculating the reference angle allows you to use the known values of trigonometric functions for these basic angles to solve more complex problems.

To find the reference angle for an angle in the second quadrant, we subtract the angle given from 180°. For example, \(180^{\circ} - 135^{\circ} = 45^{\circ}\). This means the reference angle for \(135^{\circ}\) is \(45^{\circ}\). Knowing this simplifies finding trigonometric function values like sine and cosine, as the values are the same as that of the reference angle, adjusting for the sign based on its quadrant.

Cosine Function

The cosine function measures the adjacent side over the hypotenuse in a right triangle within the unit circle, representing the x-coordinate of a point on this circle. Cosine values range from -1 to 1.\[\text{Mathematically,} \ \ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \].

Cosine is an even function, meaning that \[ \cos(-\theta) = \cos(\theta)\] except when considering the sign change from moving into a different quadrant. That’s essential because while the angle's magnitude doesn't change, the quadrant affects the sign.

In this exercise, our task was to find \( \cos(-225^{\circ})\) which upon conversion gave \(\cos(135^{\circ})\). Using our initial knowledge that a reference angle of \(45^{\circ}\) exists in the equation and recognizing the second quadrant's rules, we determined:\[\cos(135^{\circ}) = -\frac{\sqrt{2}}{2}\] This matches the negative cosine value owing to its quadrant location.