Problem 35 - Algebra and Trigonometry

Question

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\). \(\left(-\frac{5}{13},-\frac{12}{13}\right)\)

Step-by-Step Solution

Answer

\(\sin t = -\frac{12}{13}\), \(\cos t = -\frac{5}{13}\), \(\tan t = \frac{12}{5}\).

Step 1: Understanding the Terminal Point

The terminal point \(P(x, y)\) on the unit circle represents the coordinates \((\cos t, \sin t)\). In this case, we are given the point \((-\frac{5}{13}, -\frac{12}{13})\).

Step 2: Identifying \(\cos t\)

From the point \((-\frac{5}{13}, -\frac{12}{13})\), we identify \(\cos t = x = -\frac{5}{13}\).

Step 3: Identifying \(\sin t\)

From the point \((-\frac{5}{13}, -\frac{12}{13})\), we identify \(\sin t = y = -\frac{12}{13}\).

Step 4: Calculating \(\tan t\)

The tangent function is defined as \(\tan t = \frac{\sin t}{\cos t}\). Substitute the values found: \(\tan t = \frac{-\frac{12}{13}}{-\frac{5}{13}}\). Simplify this fraction to obtain \(\tan t = \frac{12}{5}\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. Every point on the unit circle can be represented by coordinates \( (x, y) \), where these coordinates relate directly to trigonometric functions.
In the context of the unit circle, the horizontal coordinate represents the cosine of an angle, and the vertical coordinate represents the sine of that angle. This is because the circle is defined by the equation \( x^2 + y^2 = 1 \), consistent with the Pythagorean identity for sine and cosine: \( \cos^2 t + \sin^2 t = 1 \).

• The angle, often denoted by \( t \), is measured from the positive x-axis.
• It can represent any real number, moving counterclockwise for positive angles and clockwise for negative angles.

This simple structure makes it easier to define and visualize the sine and cosine functions, especially when handling angles beyond the usual 0 to 360-degree range.

Sine Function

The sine function is one of the primary trigonometric functions and is vital in understanding periodic phenomena, such as sound and light waves. It maps an angle to the y-coordinate of the corresponding point on the unit circle.
When you have a point \( (x, y) \) on the unit circle, its y-value is \( \sin t \). This is evident from the unit circle's relationship to sine.

• The sine function can produce values between \(-1\) and \(1\).
• It reaches its maximum value of \(1\) at \(90^\circ\) (or \(\frac{\pi}{2}\)), and its minimum value of \(-1\) at \(270^\circ\) (or \(\frac{3\pi}{2}\)).

In this example, from the point \( (-\frac{5}{13}, -\frac{12}{13}) \), we see that \( \sin t = -\frac{12}{13} \). This represents the oscillating nature of the sine function.

Cosine Function

The cosine function is another key trigonometric function. Similar to the sine function, it is also derived from the coordinates of a point on the unit circle. It corresponds to the x-coordinate of the point.
For any angle \( t \), \( \cos t \) gives the horizontal distance from the center of the circle to the point on its circumference. It shares similar properties with the sine function:

• Its values are confined to the interval \([-1, 1]\).
• It equals \(1\) at \( t = 0^\circ \) (or \(0 \)) and \(-1\) at \( t = 180^\circ \) (or \(\pi\)).

Given the terminal point \( (-\frac{5}{13}, -\frac{12}{13}) \), it is clear that \( \cos t = -\frac{5}{13} \). This value helps determine the notion of direction, either left or right from the origin.

Tangent Function

The tangent function, often less intuitive than sine and cosine, relates closely to the slope of a line. It is defined as the ratio of the sine to the cosine of an angle.
This means for any angle \( t \), the tangent is calculated using the formula \( \tan t = \frac{\sin t}{\cos t} \). For the given point \( (-\frac{5}{13}, -\frac{12}{13}) \):

• Here, \( \tan t = \frac{-\frac{12}{13}}{-\frac{5}{13}} = \frac{12}{5} \).
• Unlike sine and cosine, tangent is not limited to any specific values and can range from \(-\infty\) to \(\infty\).

Since tangent represents a slope, a positive value indicates a line that rises both from left to right and vice versa for negative values. Understanding this aspect of the tangent function enriches our comprehension of angles and helps solve various geometric problems.