Problem 34 - Precalculus Mathematics for Calculus

Question

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\). $$\left(\frac{40}{41}, \frac{9}{41}\right)$$

Step-by-Step Solution

Answer

\(\sin t = \frac{9}{41}, \cos t = \frac{40}{41}, \tan t = \frac{9}{40}\).

Step 1: Understanding the Terminal Point

The terminal point given is \(P\left(\frac{40}{41}, \frac{9}{41}\right)\). In the unit circle, this point corresponds to coordinates \(x = \frac{40}{41}\) and \(y = \frac{9}{41}\). This means that \(x\) is \(\cos t\) and \(y\) is \(\sin t\).

Step 2: Finding \(\cos t\)

Since \(x = \frac{40}{41}\) corresponds to \(\cos t\), we have \(\cos t = \frac{40}{41}\).

Step 3: Finding \(\sin t\)

Since \(y = \frac{9}{41}\) corresponds to \(\sin t\), we have \(\sin t = \frac{9}{41}\).

Step 4: Finding \(\tan t\)

\(\tan t\) is calculated using the formula \(\tan t = \frac{\sin t}{\cos t}\). Substituting the values, \(\tan t = \frac{\frac{9}{41}}{\frac{40}{41}} = \frac{9}{40}\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry that helps us connect angles to their trigonometric functions. Picture a circle with its center at the origin of a coordinate plane and a radius of 1. This is the unit circle. Here’s why it’s significant:

• Each point on the unit circle corresponds to a specific angle, often expressed in radians.
• The x-coordinate of a point on the circle represents the cosine of the angle, while the y-coordinate represents the sine.
• The circle allows us to understand how these functions repeat and change with the angle.

The beauty of the unit circle is that it simplifies the study of trigonometric functions. Instead of memorizing countless values, you can visualize how the cosine, sine, and tangent change as you move around the circle.

Sine

In trigonometry, the sine of an angle is a basic function that measures the vertical component of an angle when placed on the unit circle. For a given point \(P(x, y)\), the sine of the real number \(t\) is equal to the y-coordinate of that point.

• From the previous exercise, with terminus point \(P(\frac{9}{41}, \frac{40}{41})\), the sine is simply the y-value.
• Hence, \( \sin t = \frac{9}{41} \).

This aspect of sine makes it easy to quickly determine using the unit circle, especially since the maximum and minimum possible values on the unit circle are -1 and 1, respectively.

Cosine

Cosine is another crucial trigonometric function that reflects the horizontal component of an angle on the unit circle. Just like with sine, for a point \(P(x, y)\), the cosine of the angle \(t\) is found by looking at the x-coordinate.

• In our example with the terminal point \(P(\frac{40}{41}, \frac{9}{41})\), the \(\cos t\) is the x-value.
• Therefore, \( \cos t = \frac{40}{41} \).

Cosine allows us to assess how much of the angle lies along the horizontal axis. Like sine, it varies between -1 and 1 as you go around the unit circle, making it easy to relate angles to points.

Tangent

Tangent is a trigonometric function which gives us a sense of the slope created by the angle in question, putting together both vertical and horizontal components. It's defined as the ratio of sine to cosine: \(\tan t = \frac{\sin t}{\cos t}\).

• This ratio helps one characterizes the steepness or direction of the line created by the angle with respect to the unit circle.
• Using the given point \(P(\frac{40}{41}, \frac{9}{41})\), the calculated tangent is \( \tan t = \frac{\frac{9}{41}}{\frac{40}{41}} = \frac{9}{40} \).

Tangent offers a distinct way of interpreting angles, often used in calculations involving ramps, slopes, or any scenario where comparing vertical to horizontal change is useful.