Problem 36 - Algebra and Trigonometry

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

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

Step 1: Identify the coordinates

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: Understand trigonometric functions values for unit circle

For a terminal point \((x, y)\) on the unit circle, \( \cos t = x \), \( \sin t = y \) and \( \tan t = \frac{y}{x} \).

Step 3: Calculate \(\cos t\)

Using the definition of cosine on the unit circle: \( \cos t = x = \frac{\sqrt{5}}{5} \).

Step 4: Calculate \(\sin t\)

Using the definition of sine on the unit circle: \( \sin t = y = \frac{2\sqrt{5}}{5} \).

Step 5: Calculate \(\tan t\)

Using the definition of tangent as \( \tan t = \frac{y}{x} \): \[ \tan t = \frac{\frac{2\sqrt{5}}{5}}{\frac{\sqrt{5}}{5}} = \frac{2\sqrt{5}}{5} \times \frac{5}{\sqrt{5}} = 2 \].

Key Concepts

Understanding the Unit Circle

The unit circle is a crucial concept in trigonometry. It is a circle with a radius of exactly 1 unit, centered at the origin of a coordinate plane. This simple geometric shape is incredibly powerful because it connects geometrical concepts with trigonometric functions. Each point on the unit circle has coordinates \((x, y)\) which can be associated with an angle \(t\) measured from the positive x-axis.

In the unit circle:

• The x-coordinate corresponds to \(\cos t\), which is the cosine of angle \(t\).
• The y-coordinate corresponds to \(\sin t\), which is the sine of angle \(t\).

This way, the unit circle not only helps us find the sine and cosine of angles, but also determine the values of these functions for specific points around the circle. The beauty of the unit circle is its symmetry and how it simplifies understanding the periodic nature of trigonometric functions.

Sine and Cosine Functions

When we talk about sine and cosine, we refer to two of the primary trigonometric functions that derive from the angles measured on the unit circle.

**Sine (\(\sin\))** represents the y-coordinate of a point on the unit circle where the radius forms an angle \(t\) with the positive x-axis. It tells us the height of the point with respect to the center.

• For a given angle \(t\), \(\sin t = y\) of the corresponding point.

**Cosine (\(\cos\))** represents the x-coordinate of that same point. It measures how far along the x-axis the point is.

• For a given angle \(t\), \(\cos t = x\) of the corresponding point.

Through these two functions, we can understand how angles and coordinates relate to each other around the circle. They help explain periodic phenomena as they repeat every \(2\pi\) radians, or 360 degrees, around the circle.

The Tangent Function and its Relation to Sine and Cosine

In trigonometry, the tangent function (\(\tan\)) is another fundamental concept tightly linked to sine and cosine.

**Tangent (\(\tan\))** is the ratio of sine to cosine of an angle \(t\), and can be visualized within the unit circle.

• Mathematically, it is defined as \(\tan t = \frac{\sin t}{\cos t}\).
• Using the coordinates of the point on the unit circle, \(\tan t = \frac{y}{x}\).

This function provides insight into the slope of the line that intersects the point \((x, y)\) and the origin. When \(\cos t\) is zero, \(\tan t\) becomes undefined because dividing by zero is not permitted. Hence, tangent functions have vertical asymptotes where cosine equals zero. This results in the distinct wave-like pattern of tangent, with values repeating every \(\pi\) radians (or 180 degrees). Understanding tangent, alongside sine and cosine, gives us a complete toolkit for exploring angles and predicting results of trigonometric expressions.