Problem 83 Find the function value using co... [FREE SOLUTION]

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Precalculus: Graphs and Models, A Right Triangle Approach

Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 8: Problem 83

Find the function value using coordinates of points on the unit circle. $$\cos \frac{\pi}{4}$$

Short Answer

Expert verified

$ \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} $

Step by step solution

Understand the Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. Any point on the unit circle can be represented as $ (\cos \, \theta, \sin \, \theta) $, where $ \theta $ is the angle formed with the positive x-axis.

Locate the Angle $ \frac{\pi}{4} $

$ \frac{\pi}{4} $ radians is an angle in the first quadrant of the unit circle. This corresponds to 45 degrees.

Recall the Coordinates for $ \frac{\pi}{4} $

The coordinates for an angle of $ \frac{\pi}{4} $ on the unit circle are $ \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) $. This is because both the cosine and sine of 45 degrees are equal.

Identify the Cosine Value

For the angle $ \frac{\pi}{4} $, the x-coordinate represents the cosine value. So, $ \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

trigonometric functions

Trigonometric functions play a crucial role in mathematics, especially when dealing with angles and their relationships. They include sine, cosine, and tangent, among others. These functions help us understand the properties of angles within triangles and circles.

For the unit circle, trigonometric functions are particularly useful because the circle has a radius of 1. By definition:

The sine of an angle gives the y-coordinate of a point on the unit circle.
The cosine of an angle gives the x-coordinate.

cosine

The cosine function is a fundamental trigonometric function. It is denoted as $ \text{cos} \theta $ and is responsible for defining the relationship between the angle and the adjacent side of a right triangle, relative to the hypotenuse.

On the unit circle, the cosine of an angle $ \theta $ is represented by the x-coordinate of the point where the terminal side of the angle intersects the circle. This means that:

For an angle of $ \frac{\pi}{4} $, the cosine value is found by looking at the x-coordinate of the corresponding point.

In the specific example of $ \frac{\pi}{4} $, the coordinates are $ \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) $. As a result: \[ \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \]

coordinates

Coordinates are used to describe the position of a point in a plane. On the unit circle, coordinates of points take the form $ (x, y) $. These are derived from the trigonometric functions of angles:

The x-coordinate corresponds to the cosine value.

The y-coordinate corresponds to the sine value.

When working with angles on the unit circle, it's essential to remember that every point $ (x, y) $ represents the cosine and sine of an angle $ \theta $. For example, at \ \theta = \frac{\pi}{4} \, the coordinates are $ \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) $. This shows that both the cosine and sine values are equal to \ \frac{\sqrt{2}}{2} \ in this case.

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91影视

Find the function value using coordinates of points on the unit circle. $$\cos \frac{\pi}{4}$$

Short Answer

Step by step solution

Understand the Unit Circle

Locate the Angle \( \frac{\pi}{4} \)

Recall the Coordinates for \( \frac{\pi}{4} \)

Identify the Cosine Value

Key Concepts

trigonometric functions

cosine

coordinates

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