Problem 23 - A Graphical Approach to Precalculus with Limits

Question

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(7,-24)$$

Step-by-Step Solution

Answer

In the 4th quadrant, \( \theta \)'s measures: \(\begin{align*} \sin(\theta) &= -\frac{24}{25}, \\ \cos(\theta) &= \frac{7}{25}, \\ \tan(\theta) &= -\frac{24}{7}, \\ \cot(\theta) &= -\frac{7}{24}, \\ \sec(\theta) &= \frac{25}{7}, \\ \csc(\theta) &= -\frac{25}{24}. \end{align*}\)

Step 1: Understand the Problem

We need to sketch an angle \( \theta \) in standard position such that it has the least positive measure and the point \((7, -24)\) is on its terminal side. In standard position, angles are drawn on the coordinate plane with the vertex at the origin and the initial side along the positive x-axis.

Step 2: Identify the Quadrant

Since the point \((7, -24)\) is located in the coordinate plane, by inspecting the signs of the coordinates: \(x = 7\) (positive) and \(y = -24\) (negative), we determine the point is in the fourth quadrant.

Step 3: Calculate the Reference Angle

The reference angle \( \alpha \) is the acute angle the terminal side makes with the x-axis. We can use the tangent function since it is defined by \( \tan(\alpha) = \left|\frac{y}{x}\right| = \left|\frac{-24}{7}\right| = \frac{24}{7}\). Thus, \( \alpha = \tan^{-1}\left(\frac{24}{7}\right) \).

Step 4: Determine the Least Positive Measure of \( \theta \)

In the fourth quadrant, the angle \( \theta \) can be found as \( \theta = 360^\circ - \alpha \) degrees, realizing \( \alpha \) through an arctan calculation.

Step 5: Determine the Hypotenuse

For trigonometric functions, we first need the hypotenuse \( r \) for the right triangle: \[ r = \sqrt{x^2 + y^2} = \sqrt{7^2 + (-24)^2} = \sqrt{49 + 576} = \sqrt{625} = 25. \]

Step 6: Find Sine and Cosine

Now we can find \( \sin(\theta) \) and \( \cos(\theta) \). Remember that in the fourth quadrant, sine is negative and cosine is positive: \[ \sin(\theta) = \frac{y}{r} = \frac{-24}{25}, \quad \cos(\theta) = \frac{x}{r} = \frac{7}{25}. \]

Step 7: Find Tangent, Cotangent, Secant, and Cosecant

Using the sine and cosine, calculate the remaining trigonometric functions: \[ \tan(\theta) = \frac{y}{x} = \frac{-24}{7}, \quad \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{7}{-24}, \] \[ \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{25}{7}, \quad \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{25}{-24}. \]

Key Concepts

Angles in Standard Position

In trigonometry, an angle is said to be in "standard position" when its vertex is located at the origin of the coordinate plane. The initial side of the angle lies along the positive x-axis. This is like starting your measurement from the horizontal line stretching to the right. The measure of the angle depends on its movement relative to the initial side:

• If the terminal side (the second side of the angle) rotates counterclockwise, the angle is considered positive.
• If it rotates clockwise, the angle is considered negative.

By ensuring that angles are consistently drawn starting from a specific point (the x-axis with vertex at the origin), it is easier to compare and reference them. The point provided, such as (7, -24), helps determine the final location of the terminal side. This angle will have its terminal side positioned such that it passes through the given point.

Reference Angle

The reference angle is crucial as it is the smallest angle (always positive and acute), the terminal side of the original angle makes with the x-axis. To find this, we analyze the absolute values of the coordinates of the given point. For instance, in the exercise, the point (7, -24) is considered:

• Calculate the tangent of the reference angle: \(\tan(\alpha) = \left| \frac{-24}{7} \right| = \frac{24}{7}\).
• From this, determine \(\alpha\) using the inverse tangent function: \(\alpha = \tan^{-1}\left(\frac{24}{7}\right)\).

By calculating the reference angle, we consistently determine the equivalent position of any angle, regardless of its size or quadrant, simplifying many trigonometric calculations and ensuring precise angle measurements.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we plot points using pairs of numbers. It consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at the origin (0,0). Here's how to navigate the plane:

• Each point on the plane is described by coordinates (x, y).
• The plane is divided into four quadrants: - Quadrant I: x > 0, y > 0 - Quadrant II: x < 0, y > 0 - Quadrant III: x < 0, y < 0 - Quadrant IV: x > 0, y < 0
• The location of a point determines the possible sign of trigonometric functions: each function has consistent signs across specific quadrants.

In this solution, the point (7,-24) lands in Quadrant IV, where cosine remains positive, but sine is negative. Thus, understanding the coordinate plane is key to predicting the behavior of trigonometric functions.

Hypotenuse Calculation

In trigonometry, especially when solving problems related to right triangles on a coordinate plane, finding the hypotenuse is essential. It represents the longest side of a right triangle, opposite the right angle. To compute this using two points in the coordinate plane, apply the Pythagorean theorem:

• Given a point (x,y), derive the hypotenuse \(r\) or the distance of the point from the origin: \( r = \sqrt{x^2 + y^2} \).
• For (7,-24), calculate:\( r = \sqrt{7^2 + (-24)^2} = \sqrt{49 + 576} = \sqrt{625} = 25\).

This calculation provides the input needed for determining trigonometric ratios like sine and cosine, which use the side lengths of the right triangle relative to this hypotenuse.