Problem 28 - Algebra and Trigonometry

Question

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\) $$ \left(-\frac{3}{5}, \frac{4}{5}\right) $$

Step-by-Step Solution

Answer

\( \sin t = \frac{4}{5}, \cos t = -\frac{3}{5}, \tan t = -\frac{4}{3} \).

Step 1: Identify terminal point

The terminal point given is \( P \left( -\frac{3}{5}, \frac{4}{5} \right) \). This means that \( x = -\frac{3}{5} \) and \( y = \frac{4}{5} \) are the coordinates of the terminal point on the unit circle.

Step 2: Calculate \( \sin t \)

In a unit circle, the \( y \)-coordinate of a terminal point is equal to \( \sin t \). Therefore, given \( y = \frac{4}{5} \), we have \( \sin t = \frac{4}{5} \).

Step 3: Calculate \( \cos t \)

In a unit circle, the \( x \)-coordinate of a terminal point is equal to \( \cos t \). Therefore, given \( x = -\frac{3}{5} \), we have \( \cos t = -\frac{3}{5} \).

Step 4: Calculate \( \tan t \)

The tangent of an angle \( t \) is the ratio of \( \sin t \) to \( \cos t \). Hence, \( \tan t = \frac{\sin t}{\cos t} = \frac{\frac{4}{5}}{-\frac{3}{5}} = -\frac{4}{3} \).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry that helps us understand the relationships between angles and trigonometric functions. Imagine a circle with a radius of one unit, centered at the origin (0,0) on a coordinate plane. This circle is called the unit circle. It forms the foundation for defining the trigonometric functions sine, cosine, and tangent.

The unit circle allows us to express angles in radians, where one complete revolution around the circle is equal to \(2\pi\) radians. Points on the unit circle correspond to angles measured from the positive x-axis in a counterclockwise direction. Each point on the unit circle is represented as \((x, y) = (\cos t, \sin t)\), where \(t\) is the angle in radians.

Understanding the coordinates on the unit circle helps in visualizing how the trigonometric functions relate to angles. This is particularly useful because any point \((x, y)\) on the circle corresponds to values of sine and cosine for a specific angle, making it a powerful tool for solving trigonometry problems.

Sine and Cosine

Sine and cosine are two primary trigonometric functions used to describe the relationship between the angles and side lengths of right triangles. On the unit circle, these functions are directly linked to the coordinates of points. The sine of an angle \(t\), denoted \(\sin t\), is the y-coordinate of the corresponding point on the unit circle, while the cosine \(\cos t\) is the x-coordinate.

In the given terminal point \( P \left( -\frac{3}{5}, \frac{4}{5} \right) \), the y-coordinate \(\frac{4}{5}\) represents \(\sin t\) and the x-coordinate \(-\frac{3}{5}\) represents \(\cos t\). Thus, we find that \(\sin t = \frac{4}{5}\) and \(\cos t = -\frac{3}{5}\).

These functions not only help in finding angles and side lengths in triangles but also play a critical role in various fields such as physics, engineering, and computer graphics. Knowing the values of sine and cosine can simplify solving complex problems involving periodic phenomena or wave patterns.

Tangent Function

The tangent function is another essential trigonometric function which is derived from sine and cosine. It is defined as the ratio of the sine of an angle to the cosine of that angle. Mathematically, the tangent function is expressed as \(\tan t = \frac{\sin t}{\cos t}\).

Considering our example with the terminal point \( P \left( -\frac{3}{5}, \frac{4}{5} \right) \), we have already determined \(\sin t = \frac{4}{5}\) and \(\cos t = -\frac{3}{5}\). Plugging these values into the tangent formula gives us \(\tan t = \frac{\frac{4}{5}}{-\frac{3}{5}} = -\frac{4}{3}\).

The tangent function is particularly useful for understanding slopes and angles of lines in coordinate geometry. Since it represents the slope of the line formed by the terminal point and the origin, it provides insights into inclination and direction. In practical applications, knowing the tangent can aid in solving problems related to angles of elevation or depression, and is also significant in calculus for dealing with derivatives and integrals related to circular motion.