Problem 51 - Algebra 2

Question

a. Use a calculator to find the value of each expression: \(\cos 40^{\circ}, \cos 400^{\circ},\) and \(\cos \left(-320^{\circ}\right) .\) b. Critical Thinking What do you notice about the values you found in part (a)? Explain.

Step-by-Step Solution

Answer

Cosine values of 40°, 400° and -320° are equivalent as they are co-terminal angles. This property of the cosine function is noticeable in the computed results.

Step 1: Calculating Cosine Values

Use a calculator to compute the cosine values of 40°, 400° and -320°. Since 360° makes a complete cycle, our calculator would compute 400° as \( \cos (400° - 360°) \) or \( \cos 40° \) and -320° as \( \cos (360° - 320°) \) or \( \cos 40° \)

Step 2: Establish Co-terminal Equivalence

The angles 40°, 400°, and -320° are either equivalent or co-terminal. Meaning, despite their differences in degrees, they land on the same point on the unit circle. Therefore, they have the same cosine values.

Step 3: Analyze and Compare

Analyze the computed cosine values. Are they equivalent? They should be since the cosine function only depends on where the terminal side of the angle intersects the unit circle, not how it got there. In this case, all three angles intersect the unit circle at the same point, and hence, their cosine values should be equivalent.

Key Concepts

Cosine Function

The cosine function is a fundamental concept in trigonometry and is vital in relating angles to side lengths in right triangles. Cosine is specifically associated with the adjacent side over the hypotenuse of a right-angled triangle. In mathematical terms, for an angle \( \theta \) in a right triangle, cosine is:

• \( \cos(\theta) = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} \)

Beyond right triangles, the cosine function extends to the unit circle, which allows it to handle any angle. It is periodic with a cycle of \( 360^{\circ} \), meaning after a full rotation, the function values repeat. Because of this periodic nature, angles like \( 40^{\circ} \), \( 400^{\circ} \), and \(-320^{\circ}\) can have the same cosine values, as they are effectively the same angle concerning the circle’s complete rotation.

Unit Circle

The unit circle is a crucial tool in trigonometry, representing angles and trigonometric functions geometrically. It's a circle with a radius of 1 centered at the origin of a coordinate plane.

• Every point on the unit circle can be described as \((\cos(\theta), \sin(\theta))\), where \(\theta\) is the angle formed with the positive x-axis.

This circle provides a visual representation of angles and helps understand why certain angles share the same sine and cosine values. Any angle measured in degrees or radians "travels" around the circle. Since the circle's circumference is \(2\pi\), named after a complete rotation, every angle can be related back to an angle between \(0^{\circ}\) and \(360^{\circ}\) or between \(0\) and \(2\pi\) radians. This relationship helps identify co-terminal angles and justifies why their trigonometric values are the same.

Co-terminal Angles

Co-terminal angles are angles that share the same terminal side when plotted in standard position on the unit circle. Despite having different measures, co-terminal angles "end" at the same point after full rotations or combinations of full and partial rotations.

• To find a co-terminal angle, you can add or subtract \(360^{\circ}\) from an angle's measure.

For example, \( -320^{\circ} \), \( 40^{\circ} \), and \( 400^{\circ} \) are co-terminal angles because they all ultimately "point" in the same direction when considered in a unit circle configuration. As such, these angles will have identical trigonometric values, such as cosine, signifying geometric equivalence on the circle.

Angle Measurement

Angle measurement is foundational in trigonometry, as it determines how we interpret the movement around a circle. Angles can be measured in degrees or radians and affect the calculation and understanding of trigonometric functions.

• Degrees are based on dividing a full circle, \(360^{\circ}\), into equal parts.
• Radians are another measurement, where a full circle is \(2\pi\) radians.

Understanding the conversion between these units is crucial for many trigonometric applications. When solving problems, recognizing these measurements helps apply concepts like co-terminal angles. For instance, by understanding 400 degrees as \(40^{\circ} + 360^{\circ}\), it becomes clear why this angle can share the same trigonometric values with 40 degrees, due to completing a full circle.