Problem 16 - Precalculus: A Concise Course

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. $$ (-4,10) $$

Step-by-Step Solution

Answer

The six trigonometric function values of the angle in the position (-4,10) are \(\sin\theta = \frac{5\sqrt{29}}{29}, \cos\theta = -\frac{2\sqrt{29}}{29}, \tan\theta = -\frac{5}{2}, \cot\theta = -\frac{2}{5}, \sec\theta = -\frac{\sqrt{29}}{2}, \csc\theta = \frac{\sqrt{29}}{5}\)

Step 1: Set Variables x and y

Firstly, assign the given values to \(x\) and \(y\) respectively. Thus, \(x = -4\) and \(y = 10\). These represent horizontal and vertical components of the angle's position in the plane.

Step 2: Calculate r (Distance from Origin)

Next, we need to find \(r\). This is the distance from the origin to the point in the plane. Use the Pythagorean identity \(r = \sqrt{x^2 + y^2}\), to get \(r = \sqrt{(-4)^2 + 10^2} = \sqrt{116}\). Therefore, \(r = 2\sqrt{29}\), by simplifying inside the square root.

Step 3: Determine Each of the Six Trigonometric Functions

With \(x\), \(y\), and \(r\) known, we can now calculate the functions:\n1) \(\sin\theta = \frac{y}{r} = \frac{10}{2\sqrt{29}} = \frac{5}{\sqrt{29}} = \frac{5\sqrt{29}}{29}\)\n2) \(\cos\theta = \frac{x}{r} = \frac{-4}{2\sqrt{29}} = -\frac{2\sqrt{29}}{29}\)\n3) \(\tan\theta = \frac{y}{x} = \frac{10}{-4} = -\frac{5}{2}\)\nThe cotangent, secant, and cosecant are reciprocals of tangent, cosine, and sine respectively, so they are:\n4) \(\cot\theta = -\frac{2}{5}\)\n5) \(\sec\theta = -\frac{\sqrt{29}}{2}\)\n6) \(\csc\theta = \frac{\sqrt{29}}{5}\)

Key Concepts

Pythagorean Identity

Understanding the Pythagorean identity is crucial when working with trigonometric functions, especially when calculating the distance of a point from the origin in a coordinate plane. The Pythagorean identity is based on a well-known theorem from geometry that states in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

When dealing with trigonometry, the identity is written as: \[\begin{equation} \sin^2\theta + \cos^2\theta = 1 \end{equation}\]This equation shows a profound relationship between the sine and cosine functions of an angle \( \theta \). In our exercise, the term \( r \) can be thought of as the hypotenuse of a right-angled triangle formed by the x-coordinate, y-coordinate, and the line between the origin and the point \((-4, 10)\). To calculate the exact trigonometric function values, we first need to find the value of \( r \) using the modified Pythagorean theorem: \[\begin{equation} r = \sqrt{x^2 + y^2} \end{equation}\]Once \( r \) is determined, the Pythagorean identity helps us in understanding the fundamental relationship that the sine and cosine values uphold, which paves the way for finding other trigonometric functions.

Standard Position Angle

A standard position angle is one whose vertex is at the origin of a coordinate system and whose initial side lies along the positive x-axis. The terminal side of the angle is determined by rotating from the initial side to a certain position defined by the coordinates of a point, in this case, \((-4, 10)\). The position of this point relative to the origin tells us which quadrant the terminal side is in and, importantly, affects the sign (positive or negative) of the trigonometric functions.

For the given point \((-4, 10)\), since the x-coordinate is negative and the y-coordinate is positive, the terminal side of the angle lies in the second quadrant. This tells us that the sine function value will be positive (since it's based on the y-coordinate), while the cosine function (based on the x-coordinate) will be negative. Understanding this concept is essential for correctly determining the exact values of the six trigonometric functions as we move onto calculating them.

Exact Trigonometric Values

In trigonometry, determining the exact values of trigonometric functions for a given angle is a fundamental skill. These are values that do not require approximation and are typically expressed in terms of radicals and rational numbers. In our exercise, calculating the exact trigonometric values begins with knowing the coordinates of a point that the angle in standard position passes through and the length of the line segment from the origin to this point, \(r\).

The sine, cosine, and tangent functions are directly related to the coordinates of our point \((-4, 10)\), and the other three trigonometric functions—cotangent, secant, and cosecant—are their reciprocals. To determine these functions exactly, we use the following formulas:

• \(\sin\theta = \frac{y}{r}\)
• \(\cos\theta = \frac{x}{r}\)
• \(\tan\theta = \frac{y}{x}\)

Once these are calculated, the reciprocals are straightforward:

• \(\cot\theta = \frac{1}{\tan\theta}\)
• \(\sec\theta = \frac{1}{\cos\theta}\)
• \(\csc\theta = \frac{1}{\sin\theta}\)

These values are 'exact' because they are simplified as much as possible but not rounded or expressed as decimals. Those calculations, as demonstrated in the solution to our exercise, continue our path to fully understand and visualize trigonometric functions and angles.