Problem 30 - 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. $$(0,-3)$$

Step-by-Step Solution

Answer

\(\theta = \frac{3\pi}{2}\); \(\sin = -1\), \(\cos = 0\), \(\tan, \sec, \cot\) undefined; \(\csc = -1\).

Step 1: Determine the Quadrant

The point \((0, -3)\) lies on the negative y-axis. In standard position, this means the angle \(\theta\) makes a straight downward line from the origin through the point \((0, -3)\). Thus, \(\theta = \frac{3\pi}{2}\) radians or 270 degrees.

Step 2: Calculate Sine and Cosine

Since the point lies on the y-axis, we can use it to find the sine and cosine. The radius (distance from the origin to the point) is 3. Therefore, \(\sin(\theta) = \frac{-3}{3} = -1\) and \(\cos(\theta) = \frac{0}{3} = 0\).

Step 3: Calculate Tangent

The tangent of \(\theta\) is given by \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{-1}{0}\), which is undefined since division by zero is not possible.

Step 4: Calculate Cosecant, Secant, and Cotangent

The cosecant of \(\theta\) is \(\csc(\theta) = \frac{1}{\sin(\theta)} = -1\). The secant is \(\sec(\theta) = \frac{1}{\cos(\theta)}\), which is undefined because \(\cos(\theta) = 0\). The cotangent is \(\cot(\theta) = \frac{1}{\tan(\theta)}\), which is also undefined for the same reason as tangent.

Key Concepts

Angles in Standard Position

In trigonometry, angles in standard position are crucial to understanding how angles relate to a coordinate system. An angle is said to be in standard position if its vertex is at the origin of a coordinate plane, its initial side lies along the positive x-axis, and the terminal side is the ray that rotates around the origin.
When sketching angles, the direction of rotation is important. A counterclockwise rotation from the initial side to the terminal side results in a positive angle, whereas a clockwise rotation gives a negative angle.
For angle \(\theta \) with terminal side intersecting a specific point, such as \((0, -3)\), the angle is found by observing its quadrant or axis location. Here, the terminal side is directly along the negative y-axis, making it a special angle of \(\frac{3\pi}{2}\) radians or 270 degrees.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are fundamental for understanding the relationships and properties of angles and triangles in the coordinate plane. These functions include:

• Sine \(\sin \theta \)
• Cosine \(\cos \theta \)
• Tangent \(\tan \theta \)
• Cosecant \(\csc \theta \)
• Secant \(\sec \theta \)
• Cotangent \(\cot \theta \)

For a standard position angle, these functions can be calculated using the coordinates of the point on its terminal side and the radius, which is the distance from the origin to this point. For the point \((0, -3)\), the sine, cosine, and tangent can be directly determined because it lands on the y-axis. Sine corresponds to the y-coordinate divided by the radius, cosine to the x-coordinate divided, and tangent is the sine divided by cosine. If cosine equals zero, tangent and some reciprocal functions are undefined.

Radians and Degrees

Radians and degrees are two units for measuring angles, and understanding the conversion between them is essential in trigonometry. One complete rotation around a circle is 360 degrees or \(2\pi\) radians.
To convert from degrees to radians, multiply the number of degrees by \(\frac{\pi}{180}\). Conversely, to convert radians to degrees, multiply by \(\frac{180}{\pi}\).
In this exercise, the angle at \((0, -3)\) is \(rac{3\pi}{2}\) radians, which directly translates to 270 degrees, illustrating how to visualize and interpret angles in both measurement systems. Knowing both units allows for flexibility and understanding in various trigonometric computations and applications.

Coordinate Plane

The coordinate plane is a two-dimensional surface determined by two axes: the horizontal x-axis and the vertical y-axis, intersecting at the origin (0,0). It is a tool used to plot points, lines, and curves representing algebraically defined relationships.
The origin serves as the starting point for angles in standard position, a consistent reference for determining their measures.

• The x-axis is where angles start their measurement.
• The y-axis is often used to determine key trigonometric values, especially when considering points like (0, -3).

Knowing how to navigate and interpret this plane helps in visualizing angles and calculating trigonometric functions effectively. Each quadrant and axis in this plane has distinct characteristics that influence the sign and value of the trigonometric functions, especially when special angles align with the axes like \(rac{3\pi}{2}\) radians lying along the negative y-axis.