Problem 38 - Algebra and Trigonometry

Question

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\). \(\left(\frac{24}{25},-\frac{7}{25}\right)\)

Step-by-Step Solution

Answer

\( \sin t = -\frac{7}{25} \), \( \cos t = \frac{24}{25} \), \( \tan t = \frac{-7}{24} \).

Step 1: Identify the Coordinates of Point P

We start by recognizing that the given terminal point \( P \) is \( \left( \frac{24}{25}, -\frac{7}{25} \right) \). This point represents the cosine and sine of the angle \( t \) respectively.

Step 2: Find \( \cos t \) and \( \sin t \) from Coordinates

The coordinates of the terminal point \( P \) give us \( \cos t = \frac{24}{25} \) and \( \sin t = -\frac{7}{25} \). These values are derived directly from the point \( P(x, y) \).

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

To find \( \tan t \), we use the identity: \( \tan t = \frac{\sin t}{\cos t} \). Substitute the known values: \( \tan t = \frac{-\frac{7}{25}}{\frac{24}{25}} \). Simplifying this gives \( \tan t = \frac{-7}{24} \).

Key Concepts

Understanding Sine

The sine function is an essential element of trigonometry, used to calculate the vertical coordinate of a point on the unit circle that corresponds to an angle. For any angle \( t \), \( \sin t \) represents the y-coordinate of the corresponding point. In simpler terms, it essentially gives us the height from the x-axis up to our point on a circle centered at the origin.
In the given exercise, the terminal point \( (x, y) = \left( \frac{24}{25}, -\frac{7}{25} \right) \) tells us that \( \sin t = -\frac{7}{25} \). This value directly represents how much the point is below the x-axis, because it is negative. Understanding sine helps us to understand how angles impact heights and depths in circles and oscillations.

Understanding Cosine

Cosine is another fundamental trigonometric function measuring the horizontal distance from the y-axis to the terminal point on the unit circle. For an angle \( t \), \( \cos t \) indicates the x-coordinate of that same point. This gives a clear view of how far the point is along the x-axis from the circle's center.
In our exercise, the terminal point \( \left( \frac{24}{25}, -\frac{7}{25} \right) \) conveys \( \cos t = \frac{24}{25} \). This shows how far the point is situated along the x-axis. Cosine values range between -1 and 1, showing us the range of horizontal displacements possible on a unit circle. Being positive, the cosine tells us the point is to the right of the y-axis.

Understanding Tangent

The tangent function connects sine and cosine in a valuable way. It describes the slope of the line that wraps around the unit circle, connecting to the point \( (x, y) \). Mathematically, this is expressed as \( \tan t = \frac{\sin t}{\cos t} \). Essentially, tangent shows the ratio of the sine of an angle to its cosine.
In the exercise, substituting \( \sin t = -\frac{7}{25} \) and \( \cos t = \frac{24}{25} \), we calculate \( \tan t = \frac{-\frac{7}{25}}{\frac{24}{25}} = \frac{-7}{24} \). Tangent can describe how steep a line is and aids in understanding angles in terms of real-world slopes and tilts. It varies significantly, providing a wide range of values, and is crucial for navigating concepts of trigonometry.