Problem 19 - Precalculus Mathematics for Calculus

Question

Find the exact value of the trigonometric function. $$\cos 570^{\circ}$$

Step-by-Step Solution

Answer

The exact value of \( \cos 570^{\circ} \) is \( -\frac{\sqrt{3}}{2} \).

Step 1: Understand the Problem

We need to find the exact value of \( \cos 570^{\circ} \). Since 570 degrees is greater than 360 degrees, we need to find a reference angle that will help us calculate the cosine value.

Step 2: Reduce the Angle

Angles greater than 360 degrees can be reduced by subtracting multiples of 360 degrees until the resulting angle is between 0 and 360 degrees. Subtract 360 from 570: \[ 570^{\circ} - 360^{\circ} = 210^{\circ} \]Therefore, 570 degrees is equivalent to 210 degrees.

Step 3: Determine the Reference Angle

The angle 210 degrees is in the third quadrant. The reference angle for 210 degrees is calculated by subtracting 180 from it:\[ 210^{\circ} - 180^{\circ} = 30^{\circ} \]Thus, the reference angle is 30 degrees.

Step 4: Find the Cosine Using the Reference Angle

In the third quadrant, cosine values are negative. The cosine of the reference angle 30 degrees is \( \sqrt{3}/2 \). So, the cosine of 210 degrees is:\[ \cos 210^{\circ} = -\frac{\sqrt{3}}{2} \]

Step 5: Conclude the Solution

Since \( \cos 570^{\circ} \) is equivalent to \( \cos 210^{\circ} \), the exact value of \( \cos 570^{\circ} \) is:\[ \cos 570^{\circ} = -\frac{\sqrt{3}}{2} \]

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions used in mathematics. It is often used to find the horizontal component of an angle within a right triangle. The cosine function is denoted as \( \cos \) and provides the ratio of the adjacent side to the hypotenuse in a right triangle. For example, with an angle \( \theta \), the cosine function can be expressed as:\[ \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]The cosine function has a periodicity of \( 360^{\circ} \) or \( 2\pi \) radians, meaning it repeats its values every full circle.

• The range of the cosine function is from -1 to 1.
• The cosine of \( 0^{\circ} \) is 1, while \( \cos(180^{\circ}) \) and \( \cos(360^{\circ}) \) both return 0.

When dealing with angles greater than \( 360^{\circ} \), like in our problem of \( \cos 570^{\circ} \), it is necessary to find an equivalent angle within the first circle or period of the cosine function by reducing the angle.

Reference Angle

A reference angle is a simplified version of an angle which helps in finding the trigonometric function values for any given angle. Reference angles are always created by subtracting a given angle's quadrant value until the angle lies between \(0^{\circ}\) and \(90^{\circ}\). This is extremely useful because the reference angle can then easily be used to find the trigonometric function values by locating it in the corresponding quadrant's rules.

• In the first quadrant, the cosine values are positive.
• In the second quadrant, the cosine values are negative.
• In the third quadrant, like our example, cosine remains negative.
• In the fourth quadrant, cosine returns to being positive.

In our exercise, after reducing \( 570^{\circ} \) to \( 210^{\circ} \), we find the reference angle as \( 30^{\circ} \), since \( 210^{\circ} - 180^{\circ} = 30^{\circ} \). The cosine of the reference angle \( 30^{\circ} \) is \( \sqrt{3}/2 \), but because \( 210^{\circ} \) lies in the third quadrant, where all cosine values are negative, we adjust to \( -\sqrt{3}/2 \).

Unit Circle

The unit circle is a crucial concept for understanding trigonometric functions. It is a circle with a radius of 1, centered at the origin of a coordinate plane. The unit circle allows us to define trigonometric functions for any angle, not just those within triangles.

• Angles are measured in radians or degrees from the positive x-axis.
• Any point on the unit circle can be represented as \((\cos \theta, \sin \theta)\).
• The x-coordinate represents the cosine of the angle, while the y-coordinate represents the sine.

By making a full rotation around the unit circle, we cover \( 360^{\circ} \) or \( 2\pi \) radians and thus can consider angles like \( 570^{\circ} \) as being beyond a full rotation. This means we need to "reset" by subtracting multiples of \( 360^{\circ} \). With this understanding, angle \( 210^{\circ} \) becomes our working angle, whose cosine value we determine using the unit circle's negative cosine rules for the third quadrant resulting in \( -\sqrt{3}/2 \).