Problem 25 - 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. $$(0,2)$$

Step-by-Step Solution

Answer

\(\theta = \frac{\pi}{2}\); \(\sin = 1\), \(\cos = 0\), \(\tan\) undefined, \(\csc = 1\), \(\sec\) undefined, \(\cot = 0\).

Step 1: Understand the Geometry

The point \((0, 2)\) lies on the y-axis, 2 units above the origin. An angle in standard position has its vertex at the origin, and its initial side on the positive x-axis. Since the terminal side is vertical and passes through \((0, 2)\), the angle \(\theta\) counterclockwise from the positive x-axis is 90 degrees or \(\frac{\pi}{2}\) radians.

Step 2: Recognize the Reference Angle

The angle \(\theta\) is already the smallest positive angle since standard position angles are measured counterclockwise starting from the positive x-axis. Here, \(\theta = \frac{\pi}{2}\) radians.

Step 3: Calculate Trigonometric Functions

For \(\theta = \frac{\pi}{2}\):- Sine function: \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{2}{2} = 1\).- Cosine function: \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{0}{2} = 0\).- Tangent function: \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{1}{0}\), which is undefined.

Step 4: Calculate Reciprocal Trigonometric Functions

- Cosecant function: \(\csc(\theta) = \frac{1}{\sin(\theta)} = 1\).- Secant function: \(\sec(\theta) = \frac{1}{\cos(\theta)}\), which is undefined since \(\cos(\theta) = 0\).- Cotangent function: \(\cot(\theta) = \frac{1}{\tan(\theta)} = 0\).

Key Concepts

Standard Position

In trigonometry, an angle is said to be in "standard position" when two specific conditions are met. First, the angle's vertex, which is the point where the two lines making up the angle intersect, is set at the origin \( (0,0) \) of a coordinate plane. Secondly, the initial side of the angle is placed along the positive x-axis.
This makes it easier to reference and calculate trigonometric functions. With the initial side fixed, the angle is then measured by rotating counterclockwise from the positive x-axis.

This setup is incredibly useful because it provides a consistent frame of reference no matter how large or small the angle is.

• For a positive angle, rotation happens counterclockwise.
• For a negative angle, rotation is in the clockwise direction.

Standard position makes it simple to understand and visualize angles in various quadrants of the coordinate system, which helps in calculating trigonometric values.

Trigonometric Functions

Trigonometric functions relate the angles of a triangle to the lengths of its sides. In the case without a triangle, like our angle \( \theta = \frac{\pi}{2} \), we rely on their definitions in terms of the unit circle. Each trigonometric function can be understood in this general framework:

• Sine \( \sin(\theta) \): Represents the y-coordinate of the point on the unit circle for an angle \( \theta \).
• Cosine \( \cos(\theta) \): Represents the x-coordinate of the same point.
• Tangent \( \tan(\theta) \): The ratio \( \frac{\sin(\theta)}{\cos(\theta)} \), highlighting its dependency on both sine and cosine values.

Furthermore, there are reciprocal functions:

• Cosecant \( \csc(\theta) \): Reciprocal of sine \( \frac{1}{\sin(\theta)} \).
• Secant \( \sec(\theta) \): Reciprocal of cosine \( \frac{1}{\cos(\theta)} \).
• Cotangent \( \cot(\theta) \): Reciprocal of tangent \( \frac{1}{\tan(\theta)} \).

For our angle of \( \frac{\pi}{2} \), notice that some functions become undefined, such as tangent and secant because they involve division by zero.

Angle Measurement

Understanding how to measure angles is crucial in trigonometry. There are two main units of angle measurement: degrees and radians.

• Degrees: A circular motion is divided into 360 equal parts, called degrees. It's a commonly used unit especially in general day-to-day situations.
• Radians: This is a more mathematical measure where one full circle is \( 2\pi \) radians. It's derived from the radius of the circle, providing a direct link between the circle's size and the angle.

Radians are often preferred in higher mathematics and various scientific fields because they simplify many mathematical expressions. For instance, in the unit circle, one can see that:

• \( 0 \) radians is equivalent to \( 0^\circ \).
• \( \pi \) radians make up \( 180^\circ \).
• \( \frac{\pi}{2} \) radians make up a right angle, equivalent to \( 90^\circ \).

The conversion between degrees and radians is straightforward: multiply by \( \frac{\pi}{180} \) to go from degrees to radians and multiply by \( \frac{180}{\pi} \) for radians to degrees. Recognizing common angle measures in both formats helps in understanding and solving trigonometry problems effectively.