Problem 36 - 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 \(\tan t\). $$\left(\frac{\sqrt{5}}{5}, \frac{2 \sqrt{5}}{5}\right)$$

Step-by-Step Solution

Answer

\(\sin t = \frac{2\sqrt{5}}{5}\), \(\cos t = \frac{\sqrt{5}}{5}\), \(\tan t = 2\).

Step 1: Identify the Components of Point P

The terminal point is given as \(P\left(\frac{\sqrt{5}}{5}, \frac{2\sqrt{5}}{5}\right)\). The \(x\)-coordinate is \(\frac{\sqrt{5}}{5}\) and the \(y\)-coordinate is \(\frac{2\sqrt{5}}{5}\).

Step 2: Determine \(\cos t\) and \(\sin t\)

For an angle \(t\), the coordinates of the terminal point \((x, y)\) on the unit circle correspond to \(\cos t=x\) and \(\sin t=y\). Thus, \(\cos t = \frac{\sqrt{5}}{5}\) and \(\sin t = \frac{2\sqrt{5}}{5}\).

Step 3: Calculate \(\tan t\)

\(\tan t\) is defined as \(\frac{\sin t}{\cos t}\). Substituting the values obtained: \(\tan t = \frac{\frac{2\sqrt{5}}{5}}{\frac{\sqrt{5}}{5}} = \frac{2\sqrt{5}}{5} \times \frac{5}{\sqrt{5}} = 2\).

Key Concepts

Understanding the 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. When dealing with angles and trigonometric functions, the unit circle helps in defining the sine, cosine, and tangent functions.
The circumference of this circle defines all possible terminal points of angles as they extend from the positive x-axis.

• The x-coordinate of a point on the unit circle is equivalent to the cosine of the angle.
• The y-coordinate corresponds to the sine of the angle.

This makes the unit circle an invaluable tool for understanding the geometric interpretation of these functions. The point \((\frac{\sqrt{5}}{5}, \frac{2\sqrt{5}}{5})\), for example, is a point on this circle.

Exploring Sine and Cosine

Sine and cosine are two important trigonometric functions that are easily understood through their relationships with the unit circle.

• Sine \(\sin t\) represents the vertical position or y-coordinate of a point on the unit circle.
• Cosine \(\cos t\) represents the horizontal position or x-coordinate.

When you know a point on the unit circle, such as \(\left(\frac{\sqrt{5}}{5}, \frac{2\sqrt{5}}{5}\right)\), you can directly determine these trigonometric values:

• \(\cos t = \frac{\sqrt{5}}{5}\)
• \(\sin t = \frac{2\sqrt{5}}{5}\)

These values give information about the angle's position around the unit circle, which is useful for solving many trigonometric problems.

Understanding Tangent Function

The tangent function (\(\tan t\)) describes the slope of the line created by a terminal point with the origin on the unit circle.
It is computed as the ratio of sine to cosine:\[\tan t = \frac{\sin t}{\cos t}\]This means if you know the sine \(\sin t\) and cosine \(\cos t\) values, you can easily find the tangent.
For example, with the point \(\left(\frac{\sqrt{5}}{5}, \frac{2\sqrt{5}}{5}\right)\), tangent becomes:\[\tan t = \frac{\frac{2\sqrt{5}}{5}}{\frac{\sqrt{5}}{5}} = 2\]

• Tangent helps in determining the angle's acute slope, often used in trigonometric equations and real-world applications like inclines or angles of elevation.