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Chapter 6: Problem 65

Find the reference angle and the exact function value if they exist. $$\cos 0^{\circ}$$

Short Answer

Expert verified
The reference angle is 0 degrees, and \cos 0^{\circ} = 1\.

Step by step solution

01

Convert Angle to Standard Position

Since the angle is already given as 0 degrees, it is in the standard position. Therefore, there鈥檚 no need for conversion.
02

Locate the Angle on the Unit Circle

0 degrees is located at the point (1, 0) on the unit circle in standard position.
03

Determine the Cosine Value

On the unit circle, the cosine of an angle is the x-coordinate of the corresponding point. For 0 degrees, the point is (1, 0), so the cosine value is 1.
04

Find the Reference Angle

Since 0 degrees is the standard position angle and falls within the first quadrant, the reference angle is 0 degrees itself.

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Key Concepts

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

Reference Angle
A reference angle is the smallest angle that the terminal side of your given angle makes with the x-axis.
It always falls within the range of 0掳 to 90掳. This concept is useful in trigonometry, especially when dealing with angles larger than 90掳 or those that fall in different quadrants.

To find the reference angle, follow these general steps:
  • If the angle is in the first quadrant (0掳 to 90掳), the reference angle is the angle itself.

  • If the angle is in the second quadrant (90掳 to 180掳), subtract the angle from 180掳.

  • If the angle is in the third quadrant (180掳 to 270掳), subtract 180掳 from the angle.

  • If the angle is in the fourth quadrant (270掳 to 360掳), subtract the angle from 360掳.

    In the given exercise, the angle is 0掳, which is already in the standard position and falls within the first quadrant. Therefore, the reference angle is 0掳 itself.
Unit Circle
The unit circle is a circle with a radius of 1 centered at the origin on the coordinate plane.
It's a fundamental tool in trigonometry as it helps define the trigonometric functions for all angles.
You can visualize the circle, and notice that each point on the circle corresponds to the cosine and sine values of an angle.

Points on the unit circle are represented as \(x, y\) coordinates, where:
  • x-coordinate represents the cosine value of the angle.
  • y-coordinate represents the sine value of the angle.
For example, at 0掳 (or 360掳), the point is (1, 0). At 90掳, the point is (0, 1), and so on.

In this exercise, the angle 0掳 lies on the positive x-axis, so the corresponding point on the unit circle is (1, 0), making the cosine value 1 and the sine value 0.
Cosine Value
The cosine of an angle is a trigonometric function that gives the x-coordinate of the point on the unit circle that corresponds to a given angle.
Mathematically, it鈥檚 often written as \(\text{cos} \theta\), where \(\theta\) is the angle.

Cosine values can range from -1 to 1. Here are some key cosine values to remember at standard angles:
  • \(\text{cos} 0^{\text{掳}} = 1\)
  • \(\text{cos} 90^{\text{掳}} = 0\)
  • \(\text{cos} 180^{\text{掳}} = -1\)
  • \(\text{cos} 270^{\text{掳}} = 0\)

In this exercise, we are asked to find the cosine of 0掳.
When we locate 0掳 on the unit circle, we see it corresponds to the point (1, 0).
This means the cosine value is 1, as the x-coordinate at this point is 1.

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