Problem 28 - 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. $$(-5,0)$$

Step-by-Step Solution

Answer

The angle in standard position is π radians, with trigonometric function values: \( \sin \theta = 0, \cos \theta = -1, \tan \theta = 0, \cot \theta = \text{undefined}, \sec \theta = -1, \csc \theta = \text{undefined}.\)

Step 1: Identify the Point on the Coordinate System

The point (-5, 0) is on the x-axis, specifically on the negative x-axis. This tells us that the terminal side of the angle θ lies right on the negative x-axis.

Step 2: Determine the Angle in Standard Position

To determine θ, note the position on the circle. An angle whose terminal side is on the negative x-axis is θ = 180° or θ = π radians. This is the least possible positive measure for the angle with its terminal side at (-5, 0).

Step 3: Calculate the Radius

Since the point (-5, 0) lies on the x-axis, the radius, or the distance from the origin to the point, is just 5. The radius is calculated as \( r = \sqrt{(-5)^2 + 0^2} = 5 \).

Step 4: Evaluate Sine and Cosine

Using the point (-5, 0) and radius 5, we find sine and cosine: \( \sin \theta = \frac{y}{r} = \frac{0}{5} = 0 \) and \( \cos \theta = \frac{x}{r} = \frac{-5}{5} = -1 \).

Step 5: Evaluate Tangent and Cotangent

Since \( \tan \theta = \frac{y}{x} = \frac{0}{-5} = 0 \),the cotangent \( \cot \theta = \frac{x}{y} = \frac{-5}{0} \) is undefined.

Step 6: Evaluate Secant and Cosecant

The secant is the reciprocal of cosine: \( \sec \theta = \frac{1}{\cos \theta} = \frac{1}{-1} = -1 \).The cosecant is the reciprocal of sine, and since \( \sin \theta = 0 \),\( \csc \theta = \frac{1}{0} \) is undefined.

Key Concepts

Standard Position

In trigonometry, an angle is said to be in "standard position" when its vertex is located at the origin (0,0) of the coordinate plane, and its initial side lies along the positive x-axis.

This terminology provides a consistent way to describe the orientation and direction of an angle's rotation. When we talk about placing an angle in this position, it helps in easily identifying and calculating the values of trigonometric functions.

Whenever you're asked to sketch an angle in standard position, you begin by drawing the positive x-axis, then measure the angle's rotation, either clockwise or counterclockwise, from there.

Terminal Side

The "terminal side" of an angle in standard position is the ray that rotates away from the initial side, indicating the direction and magnitude of the angle being measured.

For instance, if the point (-5, 0) is given, the terminal side of the angle lands on that point when measured in standard position.
This means our angle has rotated in such a way that its end has reached the negative x-axis.

The terminal side is crucial in calculating trigonometric functions, as the coordinates it reaches help in determining sine, cosine, and other trigonometric values associated with the angle.

Rationalizing Denominators

"Rationalizing denominators" is a technique used in mathematics to eliminate square roots or irrational numbers from the denominator of a fraction.
This process is important when calculating trigonometric functions, as results can often involve irrational numbers.

To rationalize a denominator, multiply the numerator and the denominator of the fraction by a value that will remove the irrational part from the denominator. For example, for a denominator of the form \(a + \sqrt{b}\), you would multiply by \(a - \sqrt{b}\) to cancel the square root.

Though in simpler trigonometric calculations like those in this exercise, we may not encounter the need frequently, knowing how to apply this method ensures a cleaner and more standard solution format.

Radians Conversion

Radians are an alternative way to measure angles, much like degrees, but they relate directly to the geometry of a circle. One complete revolution around a circle contains \(2\pi\) radians, which equates to \(360^{\circ}\).

To convert an angle from degrees to radians, use the conversion factor \(\frac{\pi}{180}\): multiply the number of degrees by this fraction. Conversely, converting radians to degrees, multiply by \(\frac{180}{\pi}\).

In the exercise, the angle was found to be \(180^{\circ}\), which directly converts to \(\pi\) radians. Understanding these conversions is essential as it allows for flexibility between different trigonometry problems and conforming to standard mathematical practices.