Problem 22 - Algebra and Trigonometry Real Mathematics, Real People

Question

The point is on the terminal side of an angle in standard position. Determine the exact values of the six trigonometric functions of the angle. $$(-24,10)$$

Step-by-Step Solution

Answer

\(\sin= \frac{5\sqrt{179}}{179}, \cos= \frac{-12\sqrt{179}}{179}, \tan= -\frac{5}{12}, \csc= \frac{179}{5}, \sec= \frac{-179}{12}, \cot= \frac{-12}{5}\)

Step 1: Identify the coordinates

The coordinates of the point on the terminal side of an angle are (-24, 10). Let's denote the x-coordinate as x and the y-coordinate as y. Here, x = -24 and y = 10.

Step 2: Calculate the radius r

Calculate the distance from the point to the origin, which we'll call r. We can use the Pythagorean theorem for this: \( r = \sqrt{x^{2} + y^{2}} \). Substituting x = -24 and y = 10, we find \( r = \sqrt{(-24)^{2} + (10)^{2}} = \sqrt{716} = 2 \sqrt{179} \).

Step 3: Determine the six trigonometric functions

We find the six trigonometric functions using their definitions:\n\- Sine function or sin is defined as \( \sin = \frac{y}{r} = \frac{10}{2\sqrt{179}} = \frac{5}{\sqrt{179}} =\frac{5\sqrt{179}}{179}\).\n\- Cosine function or cos is defined as \( \cos = \frac{x}{r} = \frac{-24}{2\sqrt{179}} = \frac{-12}{\sqrt{179}} = \frac{-12\sqrt{179}}{179}\).\n\- Tangent function or tan is defined as \( \tan = \frac{y}{x} = \frac{10}{-24} = -\frac{5}{12}\).\n\Now, the reciprocals of sin, cos, and tan are respectively the cosecant function (csc), secant function (sec), and cotangent function (cot):\n\- Cosecant function or csc is \( \csc = \frac{1}{\sin} = \frac{179}{5}\).\n\- Secant function or sec is \( \sec = \frac{1}{\cos} = \frac{-179}{12}\).\n\- Cotangent function or cot is \( \cot = \frac{1}{\tan} = \frac{-12}{5}\).

Key Concepts

Sine Function

The sine function, often denoted as \( \sin \), is a fundamental trigonometric function. This function relates the vertical side of a right triangle to the hypotenuse when the angle is placed in standard position at the origin of a coordinate plane. In simpler terms, it calculates the ratio between the opposite side of the angle and the hypotenuse.
In our example, the point \((-24, 10)\) lies on the terminal side of our angle. The y-coordinate (here \(y = 10\)) represents the vertical side of the triangle, while \(r\) (calculated as \(2 \sqrt{179}\)) is the hypotenuse.
Thus, the sine function is calculated as follows:

• \( \sin(\theta) = \frac{y}{r} = \frac{10}{2\sqrt{179}} = \frac{5\sqrt{179}}{179} \)

The sine function helps us understand angles in triangles and how they translate into the unit circle in geometry.

Cosine Function

The cosine function, noted as \( \cos \), is another critical element of trigonometry. It expresses the horizontal component of a point on the circle relative to the hypotenuse. In practical terms, it gives us the ratio of the adjacent side of a triangle to its hypotenuse.
In the given coordinate point \((-24, 10)\), the x-coordinate \(-24\) represents the adjacent side of our triangle. Again, \(r\) is the hypotenuse we've found earlier: \(2 \sqrt{179}\).
Using this information, we calculate the cosine function:

• \( \cos(\theta) = \frac{x}{r} = \frac{-24}{2\sqrt{179}} = \frac{-12\sqrt{179}}{179} \)

The cosine function is essential in determining an angle's horizontal projection and plays a critical role in circular motion and wave functions.

Tangent Function

The tangent function, commonly expressed as \( \tan \), represents the relationship between the sine and cosine functions. Specifically, it describes the ratio of a triangle's opposite side to its adjacent side.
For our angle at point \((-24, 10)\), the y-coordinate (\(10\)) acts as the opposite, while the x-coordinate (\(-24\)) is the adjacent.
We calculate the tangent function as:

• \( \tan(\theta) = \frac{y}{x} = \frac{10}{-24} = -\frac{5}{12} \)

The tangent function aids in understanding how steep a slope is when relating to angles, and it is significant in various practical applications where angle steepness is important.

Standard Position Angle

An angle in the standard position is crucial in trigonometry. It begins with its vertex at the origin in a Cartesian plane, with its initial side aligned along the positive x-axis.
When analyzing angles, particularly ones extending beyond a single quadrant, the standard position provides a simple reference. This allows for consistency across calculations involving trigonometric functions.
Given point \((-24, 10)\) on the terminal side signifies an angle extending from the positive x-axis in a counter-clockwise direction, reaching the second quadrant.
Understanding angles in the standard position helps us systematically utilize trigonometric functions to unravel complex problems in geometry.

Pythagorean Theorem

The Pythagorean theorem is central in trigonometry and geometry. It determines the relationship among the sides of a right triangle: the square of the hypotenuse equals the sum of the squares of the other two sides.
It’s expressed mathematically as \( a^{2} + b^{2} = c^{2} \), where \(c\) is the hypotenuse.
In our trigonometric functions problem, the theorem is employed to calculate \(r\), the hypotenuse for our point \((-24, 10)\):

• \( r = \sqrt{(-24)^{2} + (10)^{2}} = \sqrt{716} = 2 \sqrt{179} \)

The Pythagorean theorem provides the bridge needed to transition coordinates into trigonometric values, crucial in many geometry and physics problems.