Problem 34 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the terminal point coordinates

The terminal point is given as \(P\left(\frac{40}{41}, \frac{9}{41}\right)\), where \(x = \frac{40}{41}\) and \(y = \frac{9}{41}\).

Step 2: Recognize the trigonometric identities

The circle definition gives us \(\cos t = x\), \(\sin t = y\), and \(\tan t = \frac{y}{x}\).

Step 3: Calculate \(\sin t\) and \(\cos t\)

From the terminal point coordinates, \(\cos t = \frac{40}{41}\) and \(\sin t = \frac{9}{41}\) by directly interpreting the \(x\) and \(y\) values.

Step 4: Determine \(\tan t\) using \(y\) over \(x\)

Calculate \(\tan t\) using the formula: \(\tan t = \frac{\sin t}{\cos t} = \frac{\frac{9}{41}}{\frac{40}{41}} = \frac{9}{40}\).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in understanding the relationships between angles and sides in right triangles. They become even more versatile when extended to the unit circle.

• Sine (\(\sin\)): It measures the vertical component of the angle, specifically the \(y\)-coordinate of a point on the unit circle.
• Cosine (\(\cos\)): It measures the horizontal component, or the \(x\)-coordinate of the same point.
• Tangent (\(\tan\)): It represents the ratio between the sine and cosine, which is easily calculated using \(\tan t = \frac{\sin t}{\cos t}\).

For the terminal point \(P\left(\frac{40}{41}, \frac{9}{41}\right)\), by directly relating these coordinates to \(\sin t\) and \(\cos t\), we extract \(\sin t = \frac{9}{41}\) and \(\cos t = \frac{40}{41}\). Additionally, \(\tan t\) is derived from dividing these two components, resulting in \(\tan t = \frac{9}{40}\). Understanding these fundamental connections allows us to solve trigonometric problems effectively.

Unit Circle

The unit circle is a pivotal concept in trigonometry and helps in visualizing trigonometric functions. It is a circle with a radius of 1 centered at the origin \((0, 0)\) on a coordinate plane.

• The radius of the unit circle ensures that any terminal point can be expressed as \((\cos t, \sin t)\).
• These points help define angles measured in radians, where the angle \(t\) is the distance from the positive \(x\)-axis around the circle.
• Values on the unit circle are periodic, which means that they repeat every \(2\pi\) radians (or 360°).

For example, a known point \(P\left(\frac{40}{41}, \frac{9}{41}\right)\) lies on the unit circle, giving us direct access to the values of \(\cos t\) and \(\sin t\). The unit circle framework simplifies this extraction by ensuring that each coordinate of \(P\) represents a basic trigonometric function's output.

Terminal Point

Understanding the concept of a terminal point is essential in trigonometry. The terminal point of a real number \(t\) is the location on the unit circle reached after traveling \(t\) radians from the positive \(x\)-axis.

• Each terminal point, given by coordinates \((x, y)\), equates to \((\cos t, \sin t)\).
• They define the sine and cosine of an angle and are used to find the tangent through \(\frac{y}{x}\).
• The terminal point helps in determining the exact values of trigonometric functions at specific angles.

In our example, the point \(P\left(\frac{40}{41}, \frac{9}{41}\right)\) serves as such a terminal point, providing key information about the angle in question. This coordinate allows direct computation of trigonometric functions by interpreting each part, \(x\), and \(y\), as \(\cos t\) and \(\sin t\), respectively, which are integral to understanding trigonometric circles.