Problem 22 - Precalculus Mathematics for Calculus

Question

Find the exact value of the trigonometric function. $$\cos 660^{\circ}$$

Step-by-Step Solution

Answer

The exact value of \(\cos 660^{\circ}\) is \(\frac{1}{2}\).

Step 1: Determine Principal Angle

The first step is to reduce the angle \(660^{\circ}\) to a principal angle between \(0^{\circ}\) and \(360^{\circ}\). To do this, subtract \(360^{\circ}\) from \(660^{\circ}\) until the result is in the desired range. \[ 660^{\circ} - 360^{\circ} = 300^{\circ} \] Thus, \(660^{\circ}\) corresponds to \(300^{\circ}\) in the standard position.

Step 2: Identify Reference Angle

To find the reference angle for \(300^{\circ}\), subtract \(300^{\circ}\) from \(360^{\circ}\). \[ 360^{\circ} - 300^{\circ} = 60^{\circ} \] The reference angle is \(60^{\circ}\).

Step 3: Use Unit Circle Values

The reference angle is in the fourth quadrant, where the cosine function is positive. The cosine of \(60^{\circ}\) is \(\frac{1}{2}\). Therefore, in the fourth quadrant, \(\cos(300^{\circ}) = \cos(60^{\circ}) = \frac{1}{2}\).

Step 4: Conclusion

Thus, the value of \(\cos 660^{\circ}\) is the same as \(\cos 300^{\circ}\), which is \(\frac{1}{2}\).

Key Concepts

Principal Angle

When working with angles in trigonometry, especially those beyond the basic 0° to 360° range, the principal angle is crucial. A principal angle is the simplest form of an angle, reduced to a value between 0° and 360°. This simplification is helpful for evaluating trigonometric functions without the complexity of large angle degrees.

To turn any given angle into its principal angle:

• Subtract 360° repeatedly until the angle falls within the 0° to 360° range.
• This process ensures the angle corresponds to a point on the unit circle, making calculations easier.

For instance, for the angle 660°, subtracting 360° gives us 300°, which is the principal angle. The process is simple yet powerful, enabling a straightforward approach to advanced trigonometric problems.

Reference Angle

A reference angle helps to simplify analysis of trigonometric functions by relating an angle to its acute equivalent measuring between 0° and 90°. This equivalency is beneficial as the trigonometric values at a reference angle can be reflected in other quadrants with simple sign changes.

To find a reference angle:

• Identify the quadrant where the angle's terminal side resides.
• If the angle is in the third or fourth quadrant, subtract the value from 360° (e.g., 360° - 300° = 60° in our case).

The 60° obtained is the reference angle for 300°, making it easier to use known trigonometric values. Remember, different quadrants will affect the sign of the trigonometric result, so always keep the sign convention in mind based on the quadrant position.

Unit Circle

The unit circle is a fundamental concept in trigonometry that represents all the angles and their trigonometric values within a circle of radius 1 centered at the origin of the coordinate plane. It is invaluable for understanding the sine, cosine, and tangent functions from -1 to 1.

Important features of the unit circle include:

• Radius equals 1, simplifying calculations and providing clear interpretations for angles.
• Cosine values correspond to the x-coordinates, and sine values correspond to y-coordinates of the point where an angle's terminal side intersects the circle.

For the angle 300°, when mapped on the unit circle, lands in the fourth quadrant. The cosine value here is positive, which is consistent with our calculation of \(\cos(300°) = \frac{1}{2}\). Using the unit circle helps visualize and verify the trigonometric values and enhances comprehension of how angles behave within the circle framework.