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Chapter 6: Problem 14

Find the exact value of the trigonometric function. $$ \cos \left(-60^{\circ}\right) $$

Short Answer

Expert verified
\( \cos(-60^{\circ}) = \frac{1}{2} \).

Step by step solution

01

Identify Trigonometric Identity

We start by using the trigonometric identity for cosine of negative angles. Using the identity: \( \cos(-\theta) = \cos(\theta) \), we can simplify our problem to \( \cos(-60^{\circ}) = \cos(60^{\circ}) \).
02

Recall Standard Value

Recall that \( \cos(60^{\circ}) \) is a standard angle in trigonometry. From the unit circle or the known values, \( \cos(60^{\circ}) \) is equal to \( \frac{1}{2} \).
03

Verify the Result Using the Unit Circle

To verify, imagine the position of 60 degrees on the unit circle. It is located in the first quadrant where cosine values are positive, confirming that \( \cos(60^{\circ}) = \frac{1}{2} \).

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Key Concepts

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

Cosine of Angles
The trigonometric function cosine, often abbreviated as cos, is a fundamental function used in many areas of mathematics. For any angle \( \theta \), the cosine of that angle represents the x-coordinate of a point on the unit circle.

The cosine value can be determined for any angle, whether it is in degrees or radians.
  • For angles in the first quadrant (0掳 to 90掳), cosine values are positive.
  • For angles in the second quadrant (90掳 to 180掳), cosine values are negative.
  • In the third quadrant (180掳 to 270掳), cosine values remain negative.
  • For the fourth quadrant (270掳 to 360掳), cosine values turn positive again.
A standard angle, like 60掳, is one that has a widely recognized cosine value. Such angles are easy to recall, facilitating quick calculations.
Negative Angle Identity
The negative angle identity for cosine makes it easier to work with angles measuring less than zero. The equation is:
\[ \cos(-\theta) = \cos(\theta) \]This simplification indicates that negative angles produce the same cosine value as their positive counterparts.

This is because cosine is an even function, meaning its graph is symmetric about the y-axis. Therefore, whether you rotate clockwise or counterclockwise, the horizontal distance (or x-coordinate) from the origin will be identical. This property is useful for solving problems that involve negative angles, as it simplifies calculations and reduces computational demands by converting it to a positive angle equivalent.
Unit Circle
The unit circle is a critical concept in trigonometry, serving as a visual aid for understanding trigonometric functions. The circle has a radius of one unit, centered at the origin (0,0) of the coordinate plane.

Each point on the unit circle corresponds to an angle \( \theta \), measured from the positive x-axis.
  • The coordinates of any point \( (x, y) \) on the unit circle are standardized as \( (\cos(\theta), \sin(\theta)) \).
  • This means for an angle of 60掳, the point will be \( \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right) \), confirming \( \cos(60^{\circ}) = \frac{1}{2} \).
The unit circle assists in quickly identifying trigonometric values, especially with angles such as 30掳, 60掳, and 90掳, due to their symmetrical properties.

Utilizing the unit circle can enhance understanding and facilitate mastery over trigonometric functions and their corresponding values.

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Most popular questions from this chapter

\(19-28\) . Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. $$ a=30, \quad c=40, \quad \angle A=37^{\circ} $$

Rewrite the expression as an algebraic expression in x. $$ \sin \left(\tan ^{-1} x\right) $$

Rewrite the expression as an algebraic expression in x. $$ \cos \left(\tan ^{-1} x\right) $$

Rewrite the expression as an algebraic expression in x. $$ \cos \left(\sin ^{-1} x\right) $$

\(19-28\) . Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. $$ a=26, \quad c=15, \quad \angle C=29^{\circ} $$
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