Problem 4 - A Graphical Approach to Precalculus with Limits

Question

How would the six trigonometric functions of \(90^{\circ}\) and \(-270^{\circ}\) compare? Why?

Step-by-Step Solution

Answer

The functions for both angles are the same because \(-270^{\circ} = 90^{\circ}\) when adding 360°.

Step 1: Understand the Angle Relation

First, note the equivalence of the angles: \(90^{\circ}\) and \(-270^{\circ}\). To find if they coincide, add \(360^{\circ}\) to \(-270^{\circ}\): \(-270^{\circ} + 360^{\circ} = 90^{\circ}\). Thus, \(90^{\circ}\) and \(-270^{\circ}\) refer to the same position on the unit circle.

Step 2: Calculate Trigonometric Functions for 90°

For the angle \(90^{\circ}\) or \(\frac{\pi}{2}\) radians on the unit circle:\(\sin 90^{\circ} = 1\)\(\cos 90^{\circ} = 0\)\(\tan 90^{\circ} = \text{undefined (since you divide by zero)}\)\(\csc 90^{\circ} = 1\)\(\sec 90^{\circ} = \text{undefined (since you divide by zero)}\)\(\cot 90^{\circ} = 0\).

Step 3: Relate Functions for -270°

Because \(-270^{\circ}\) is equivalent to \(90^{\circ}\), the trigonometric functions are the same for both angles:\(\sin (-270^{\circ}) = 1\)\(\cos (-270^{\circ}) = 0\)\(\tan (-270^{\circ}) = \text{undefined}\)\(\csc (-270^{\circ}) = 1\)\(\sec (-270^{\circ}) = \text{undefined}\)\(\cot (-270^{\circ}) = 0\).

Key Concepts

Understanding the Unit Circle

The unit circle is a fundamental tool in trigonometry. It's simply a circle centered at the origin of a coordinate plane, with a radius of one unit. This allows us to conveniently calculate trigonometric functions for any angle by relying on the coordinates of points on the circle.

Key aspects of the unit circle include:

• The x-coordinate represents the cosine of the angle.
• The y-coordinate represents the sine of the angle.
• Angles are usually measured from the positive x-axis.

Rotating counterclockwise corresponds to positive angles, while clockwise rotation gives negative angles. Because the unit circle repeats every 360°, angles like 90° and -270° are equivalent, as they both point to the same spot on the circle.

Angle Equivalence

Angle equivalence is a concept where angles that differ by multiples of 360° lead to the same trigonometric function values. This is because a full rotation around the circle doesn't change the position of points, so the trigonometric quantities remain constant.

For instance, 90° and -270° are equivalent because:

• Add 360° to -270°: \(-270° + 360° = 90°\)
• They both describe the same position on the circle at \(\frac{\pi}{2}\) radians.

Understanding this principle helps simplify calculations and recognize that angles separated by full circle rotations behave identically in trigonometric contexts.

Exploring Undefined Values

Trigonometric functions sometimes become undefined, notably when involving division by zero. In the context of our examples, the \(\tan\) and \(\sec\) functions become undefined at specific angles.

Consider \(90^{\circ}\):

• \(\tan 90^{\circ},\) equivalent to dividing the sine by the cosine, is undefined because \(\cos 90^{\circ} = 0\).
• \(\sec 90^{\circ},\) evaluated as \(1/\cos 90^{\circ},\) is also undefined.

These undefined values imply that certain trigonometric ratios cannot be computed at these angles, revealing the importance of understanding angle-specific behaviors within trigonometric functions.

Sine and Cosine Functions

The sine and cosine functions play central roles in the unit circle, directly relating to its coordinates. The sine function outputs the y-coordinate, while the cosine corresponds to the x-coordinate of any point on the circle.

For example, at \(90^{\circ}\) or \(-270^{\circ}\):

• \(\sin 90^{\circ} = 1\), meaning the point is at the top of the circle, with maximum height.
• \(\cos 90^{\circ} = 0\), indicating the point lies directly above the center on the circle's edge.

These behaviors underscore the stability of sine and cosine across equivalent angles and help visually understand angle properties such as amplitude and periodicity.