Problem 10 - Precalculus: Functions and Graphs

Question

Let \(P\) be the point on the unit circle \(U\) that corresponds to \(t .\) Find the coordinates of \(P\) and the exact values of the trigonometric functions of \(t,\) whenever possible. (a) \(-\pi\) (b) \(6 \pi\)

Step-by-Step Solution

Answer

a) Coordinates: (-1, 0); \( \cos(-\pi) = -1 \), \( \sin(-\pi) = 0 \), \( \tan(-\pi) = 0 \). b) Coordinates: (1, 0); \( \cos(6\pi) = 1 \), \( \sin(6\pi) = 0 \), \( \tan(6\pi) = 0 \).

Step 1: Understand the Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin (0,0) on the coordinate plane. An angle \( t \) in radians on the unit circle corresponds to a point \( P(x, y) \) where the coordinates represent the cosine and sine of the angle, respectively: \( x = \cos(t) \) and \( y = \sin(t) \).

Step 2: Analyze Part (a) \( t = -\pi \)

The angle \( -\pi \) corresponds to half a full circle in the clockwise direction. On the unit circle, this places us at the point \((-1, 0)\). Here, \( \cos(-\pi) = -1 \) and \( \sin(-\pi) = 0 \).

Step 3: Calculate Trigonometric Functions for \( t = -\pi \)

For the angle \( -\pi \):- \( \cos(-\pi) = -1 \)- \( \sin(-\pi) = 0 \)The tangent is undefined where cosine is zero, but here it's \( \frac{0}{-1} = 0 \). Hence:- \( \tan(-\pi) = 0 \).

Step 4: Analyze Part (b) \( t = 6\pi \)

The angle \( 6\pi \) is equivalent to 3 full circles as its equivalence mod \( 2\pi \) is 0. So, after completing 3 full rotations, we land back at the starting point of the unit circle at the point \( (1, 0) \).

Step 5: Calculate Trigonometric Functions for \( t = 6\pi \)

For the angle \( 6\pi \):- \( \cos(6\pi) = 1 \)- \( \sin(6\pi) = 0 \)And the tangent is also \( \tan(6\pi) = \frac{0}{1} = 0 \).

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical tools that relate angles in a circle to the coordinates of points on that circle. They are essential for understanding the relationship between angles and distances, as they provide a way to calculate angles' sine, cosine, and tangent. On the unit circle, these functions help us determine points based on a given angle.

In the case of trigonometric functions:

• Sine (\( \sin \theta \)) measures the vertical position of a point on the unit circle and it is determined by the y-coordinate.
• Cosine (\( \cos \theta \)) measures the horizontal position of a point and it is determined by the x-coordinate.
• Tangent (\( \tan \theta = \frac{\sin \theta}{\cos \theta} \)) is the ratio of sine to cosine, representing the slope of the line created by the angle.

The angle's position on the unit circle determines the exact values of these trigonometric functions, useful for solving various mathematical problems.

Radians

Radians are a unit of angular measure used in mathematics. Unlike degrees, which divide a circle into 360 parts, radians deal with the distance around the circle in terms of the circle's radius. Essentially, one radian is the angle created when you take a radius and lay it along the circle's circumference.

A full circle is equal to \( 2\pi \) radians or \( 360^{\circ} \). Thus:

• \( \pi \) radians make up half a circle or \( 180^{\circ} \).
• \( \frac{\pi}{2} \) radians equals a quarter circle or \( 90^{\circ} \).

This system simplifies many mathematical expressions and calculations, especially those involving the unit circle and trigonometric functions.

Cosine and Sine

Cosine and sine are foundational elements of trigonometry. On the unit circle, they give us the x and y coordinates of a point associated with a specific angle. Let's dig deeper into these:

• Cosine: The cosine of an angle \( t \) is the x-coordinate of the point on the unit circle at that angle. For example, when \( t = 0 \), \( \cos(0) = 1 \), since we start at the point (1, 0).
• Sine: Similarly, the sine of the angle \( t \) is the y-coordinate. Using the same \( t = 0 \) example, \( \sin(0) = 0 \), positioning us at (1, 0) vertically.

These functions help us understand movements and changes as angles rotate around the circle. They are symmetrical, periodic, and essential in various fields, such as physics and engineering.

Tangent Function

The tangent function is perhaps the most intriguing of the primary trigonometric functions. It represents the slope of the line created by the angle in standard position. When you are on the unit circle, tangent has particular properties:

• The formula for the tangent is the ratio \( \tan(t) = \frac{\sin(t)}{\cos(t)} \).
• If \( \cos(t) = 0 \) (like at \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \)), tangent becomes undefined due to division by zero.
• It repeats every \( \pi \) radians, compared to \( 2\pi \) for sine and cosine.

Tangent offers a different perspective on trigonometry with its logical connection to slope and ratio, expanding our understanding of angles and their relationships.