Problem 29 - 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 \(\tan t\). $$\left(\frac{3}{5}, \frac{4}{5}\right)$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Terminal Point Coordinates

The terminal point given is \(P(x, y) = \left(\frac{3}{5}, \frac{4}{5}\right)\). From this, we identify \(x = \frac{3}{5}\) and \(y = \frac{4}{5}\).

Step 2: Determine \(\sin t\)

The sine of the angle \(t\) is the ratio of the \(y\)-coordinate to the hypotenuse of the unit circle. Since we assume this point is on the unit circle, the hypotenuse is 1. Thus, \(\sin t = y = \frac{4}{5}\).

Step 3: Determine \(\cos t\)

The cosine of the angle \(t\) is the ratio of the \(x\)-coordinate to the hypotenuse of the unit circle. Hence, \(\cos t = x = \frac{3}{5}\).

Step 4: Determine \(\tan t\)

The tangent of the angle \(t\) is the ratio of the \(y\)-coordinate to the \(x\)-coordinate. Therefore, \(\tan t = \frac{y}{x} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3}\).

Key Concepts

Understanding the Sine Function

The sine function is a critical concept in trigonometry that relates to the unit circle, which has a radius of one.
In the context of a point \(P(x, y)\) on the unit circle, \( \sin t \) is defined as the \(y\)-coordinate of that point.
In other words, if you have a point \(P(x, y) = \left(\frac{3}{5}, \frac{4}{5}\right)\), the sine of the angle \(t\) would simply be the \(y\)-value, which is \(\frac{4}{5}\).

• The sine function indicates the vertical distance of the point \(P(x,y)\) from the origin.
• It ranges between -1 and 1 as it represents coordinates on the unit circle.

Understanding this concept is foundational as it connects angles with linear distances in a circular manner.

Exploring the Cosine Function

Similarly to the sine function, the cosine function is also fundamentally linked to the unit circle.
The cosine of an angle \( t \) is the \(x \)-coordinate of the point on the unit circle determined by that angle.
In our example of the terminal point \(P(x, y) = \left(\frac{3}{5}, \frac{4}{5}\right)\), the \(x\)-value, or cosine, is \(\frac{3}{5}\).

• Cosine represents how far along the horizontal axis the point \(P(x,y)\) is from the origin.
• Just like the sine function, cosine values are bounded between -1 and 1.

The cosine function helps describe the horizontal aspect of an angle's terminal point on the unit circle.

Diving into the Tangent Function

The tangent function is a bit different from sine and cosine as it represents a ratio rather than just a coordinate.
The tangent of an angle \( t \) is calculated by dividing the \(y \)-coordinate by the \(x \)-coordinate at the terminal point on the unit circle.
For our terminal point \(P(x, y) = \left(\frac{3}{5}, \frac{4}{5}\right)\), the tangent is \(\tan t = \frac{4}{3}\).

• The tangent function informs us of the steepness or inclination of the line connecting the origin to \(P(x,y)\).
• While it doesn't have a fixed range as sine and cosine, it is undefined when \( x = 0 \) since you cannot divide by zero.

Thus, tangent provides a measure of directionality and slope for angles based on their position on the unit circle.