Problem 32 - 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

The terminal point provided is \( P(x, y) = \left( \frac{40}{41}, \frac{9}{41} \right) \). This point lies on the unit circle since \( x^2 + y^2 = 1 \) for points on the circle.

Step 2: Calculate \\sin t (Sine of t)

Since the terminal point has coordinates \((x, y)\) and represents \(( \cos t, \sin t )\), the sine of \(t\) is given by the \(y\)-coordinate. Thus, \( \sin t = \frac{9}{41} \).

Step 3: Calculate \\cos t (Cosine of t)

Similarly, the cosine of \(t\) is represented by the \(x\)-coordinate of the terminal point, so \( \cos t = \frac{40}{41} \).

Step 4: Calculate \\tan t (Tangent of t)

The tangent of \(t\) is the ratio of the sine and cosine values, i.e., \( \tan t = \frac{ \sin t }{ \cos t } = \frac{ \frac{9}{41}}{ \frac{40}{41} } = \frac{9}{40} \).

Key Concepts

Understanding Sine and Cosine

When we talk about the sine and cosine functions in trigonometry, we're referring to their positions on the unit circle. This is a fundamental concept in understanding circular and trigonometric functions.
The unit circle is a circle with a radius of 1 unit was referred to determine trigonometric values. The point where the angle terminates on the circle gives us the values of sine and cosine.
- **Sine (\(\sin t\))**: In the unit circle, the sine of an angle \(t\) is the \(y\)-coordinate of the terminal point. It's all about measuring how far up or down the point is from the origin along the vertical axis.- **Cosine (\(\cos t\))**: Similarly, the cosine of an angle \(t\) is the \(x\)-coordinate of the terminal point. It tells us how far left or right the terminal point is from the origin along the horizontal axis.
For the given terminal point \(P(x, y) = \left( \frac{40}{41}, \frac{9}{41} \right)\), we find that:

• \(\sin t = \frac{9}{41}\)
• \(\cos t = \frac{40}{41}\)

These values follow from simply observing the ordered pair \((x, y)\) of the point on the unit circle.

Exploring the Unit Circle

The unit circle is a pivotal tool in trigonometry that helps us understand angles and their corresponding trigonometric values. With a perfect radius of 1, it allows for simple calculations and representations.
Every point \((x, y)\) on the unit circle satisfies the equation:\[ x^2 + y^2 = 1 \]This means if you square both the \(x\) and \(y\) coordinates and add them together, you always get 1. This property stems from the fact that all points on the unit circle are 1 unit away from the origin, hence the name "unit" circle.
The angle \(t\) in radians measures the rotation from the positive \(x\)-axis to the point\((x, y)\), and the coordinates directly give us \(\cos t\) and \(\sin t\). The terminal point \(\left(\frac{40}{41}, \frac{9}{41}\right)\) fits this perfectly as:

• \(\left(\frac{40}{41}\right)^2 + \left(\frac{9}{41}\right)^2 = 1\)

This verifies that the point lies correctly on the unit circle, allowing us to compute trigonometric ratios.

Calculating Tangent from Sine and Cosine

The tangent function in trigonometry gives us a measure of the angle's slope when viewed from the unit circle. It is calculated as the ratio of the sine and cosine of the angle:\[\tan t = \frac{ \sin t }{ \cos t }\]
In our exploration with the terminal point \(P(x, y) = \left(\frac{40}{41}, \frac{9}{41}\right)\), we already know \(\sin t = \frac{9}{41}\) and \(\cos t = \frac{40}{41}\). Thus, the tangent of \(t\) becomes:\[\tan t = \frac{ \frac{9}{41} }{ \frac{40}{41} } = \frac{9}{40}\]
This result tells us how steep the angle's corresponding line is if it were to be drawn through the point. Essentially, tangent serves to express the alignment of the angle in the circular path. This concept is especially useful in applications where you need angle inclinations or gradient calculations.