Problem 14 - Contemporary Precalculus

Question

In Exercises \(11-14\), assume that the terminal side of an angle of \(t\) radians passes through the given point on the unit circle. Find \(\sin t, \cos t,\) tan \(t\) $$(.6,-.8)$$

Step-by-Step Solution

Answer

Answer: For the given angle t radians, the sine, cosine, and tangent functions are as follows: \(\sin t = -0.8\) \(\cos t = 0.6\) \(\tan t = -\frac{4}{3}\).

Step 1: Identify the Coordinates

Given, \((0.6, -0.8)\) are the coordinates on the unit circle. Hence, x coordinate (horizontal) and y coordinate (vertical) are represented as the following: \(x = 0.6\) \(y = -0.8\).

Step 2: Find the Value of \(\sin t\) and \(\cos t\)

In the unit circle, we can find \(\sin t\) and \(\cos t\) by using the coordinates. The \(\cos t\) and \(\sin t\) are equal to the x and y coordinates of the point respectively, hence, \(\cos t = 0.6\) \(\sin t = -0.8\)

Step 3: Find the Value of \(\tan t\)

\(\tan t\) can be found by the relationship \(\tan t = \frac{\sin t}{\cos t}\) So, \(\tan t = \frac{-0.8}{0.6} = -\frac{4}{3}\) The sine, cosine, and tangent functions for the angle t radians are as follows: \(\sin t = -0.8\) \(\cos t = 0.6\) \(\tan t = -\frac{4}{3}\)

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. This circle is crucial because it helps define trigonometric functions for all angles.
When we think about angles in radians on the unit circle, the angle corresponds to a specific point on the circle's edge.
Here are some key aspects of the unit circle:

• The circumference of the unit circle is exactly equal to 2π, since the radius is 1.
• Coordinates on the unit circle represent specific angles. For instance, the angle at (0.6, -0.8) corresponds to the angle in the problem.
• The coordinates of any point on the unit circle can be used to find the associated sine and cosine values. Specifically, if the point is (x, y), then (x) represents cos(θ) and (y) represents sin(θ).
• This circle allows for the easy calculation of angle measures and the values of trigonometric functions without the need for a calculator.

Understanding the unit circle is key to mastering trigonometry, as it provides a simple geometric method to visualize functions like sine and cosine.

Sine Function

The sine function is one of the principal trigonometric functions. On the unit circle, the value of the sine function for an angle t is represented as the y-coordinate of the point where the terminal side of the angle intersects the circle.
Using the unit circle:

• When you have a point such as (0.6, -0.8), the y-coordinate (-0.8) is equivalent to ext{sin}( heta).
• This means that ext{sin}( heta) = -0.8.

The sine function is periodic, with a period of 2 ext{π}, which means it repeats its values every 2 ext{π} radians.
Understanding the relationship between the unit circle and the sine function can help students visualize wave patterns and motion along a circle, which are common in physics and engineering applications.

Cosine Function

Much like the sine function, the cosine function is another core component of trigonometry. The value of the cosine function for an angle t is represented as the x-coordinate of the point on the unit circle.

• In the provided example, the x-coordinate is 0.6, which means ext{cos}( heta) = 0.6.
• This value tells us about the horizontal displacement from the origin to the position on the unit circle, corresponding to the angle.

The cosine function also has a periodicity of 2 ext{π}, repeating every 2 ext{π} radians.
The cosine function is crucial in defining the horizontal component of movements or forces and extends into application areas such as oscillations and alternating current circuits. The interplay between sine and cosine, as visualized on the unit circle, contributes to a fundamental understanding of circular motion.

Tangent Function

The tangent function, which is a bit different from sine and cosine, represents the ratio of the y-coordinate to the x-coordinate of a point on the unit circle.
The formula for ext{tan}( heta) is: ext{tan}( heta) = \(\frac{\sin(\theta)}{\cos(\theta)}\).
In our example with point (0.6, -0.8):

• ext{tan}( heta) = \(\frac{-0.8}{0.6} = -\frac{4}{3}\).

This value can tell you about the slope of the line formed by the radius at that particular angle.
The tangent function has a period of π, which indicates it repeats every π radians. It helps describe phenomena where ratios and rates of change are relevant, such as inclines and projectile motion.
Understanding how to find the tangent value using sine and cosine can simplify solving many real-world problems involving angles and slopes.