Problem 29 - 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. $$(0,-4)$$

Step-by-Step Solution

Answer

Angle \(\theta\) is \(270^\circ\) with: \(\sin \theta = -1\), \(\cos \theta = 0\), \(\tan \theta\) undefined, \(\csc \theta = -1\), \(\sec \theta\) undefined, \(\cot \theta = 0\).

Step 1: Visualize the Point

The point (0, -4) lies on the y-axis, specifically 4 units down from the origin. This indicates that the terminal side of the angle \( \theta \) in standard position ends at this point.

Step 2: Determine the Angle

Since the point lies directly below the origin on the y-axis, the angle \( \theta \) is \( 270^\circ \) (or \( \frac{3\pi}{2} \) radians). This is the smallest positive measure of \( \theta \).

Step 3: Calculate the Hypotenuse

For the coordinates \((0, -4)\), the hypotenuse \( r \) (distance from the origin to the point) is computed as \( r = \sqrt{0^2 + (-4)^2} = 4 \).

Step 4: Calculate Sine

\( \sin \theta = \frac{y}{r} = \frac{-4}{4} = -1 \).

Step 5: Calculate Cosine

\( \cos \theta = \frac{x}{r} = \frac{0}{4} = 0 \).

Step 6: Calculate Tangent

\( \tan \theta = \frac{y}{x} = \frac{-4}{0} \). Tangent is undefined because division by zero is not permitted.

Step 7: Calculate Cosecant

\( \csc \theta = \frac{r}{y} = \frac{4}{-4} = -1 \).

Step 8: Calculate Secant

\( \sec \theta = \frac{r}{x} = \frac{4}{0} \). Secant is undefined for the same reason tangent is undefined.

Step 9: Calculate Cotangent

\( \cot \theta = \frac{x}{y} = \frac{0}{-4} = 0 \).

Key Concepts

Standard Position

When you're learning about angles in trigonometry, you'll often come across the term "standard position." What does this mean? An angle is in standard position if its vertex is at the origin of a coordinate plane. Additionally, the initial side of the angle lies on the positive x-axis. The other ray, known as the terminal side, is where the angle opens up.
In our exercise, there's a specific point, (0, -4), on the terminal side of the angle \(\theta\). Since this point is directly along the y-axis, we can easily identify that \(\theta\) is a vertically downward angle.

• The initial side is along the positive x-axis.
• The terminal side ends straight down on the y-axis at (0, -4).

This configuration gives an angle of \(270^\circ\) or \(\frac{3\pi}{2}\) radians. It's important to visualize and understand how the angle grows out from the x-axis. This helps in sketching angles and interpreting their standard positions.

Trigonometric Functions

Trigonometric functions allow us to understand relationships within a right triangle and extend those ideas to any angle. They are based on ratios of the sides of a triangle. In trigonometry, we often reference six key functions: sine, cosine, tangent, cosecant, secant, and cotangent.
Let's compute these for angle \(\theta = 270^\circ\) where the point (0, -4) is on the terminal side:

• Sine \(\sin \theta\): This is the ratio of the y-value to the hypotenuse, given by \(\frac{-4}{4} = -1\).
• Cosine \(\cos \theta\): This ratio reviews the link between x-value and hypotenuse, yielding \(\frac{0}{4} = 0\).
• Tangent \(\tan \theta\): Defined as \(\frac{y}{x}\), it is \(\frac{-4}{0}\), which means tangent is undefined here due to division by zero.
• Cosecant \(\csc \theta\): Reciprocal of sine, \(\frac{r}{y} = -1\).
• Secant \(\sec \theta\): It's the reciprocal of cosine, but because cosine is 0, secant is \(\frac{4}{0}\), hence undefined.
• Cotangent \(\cot \theta\): This is the reciprocal of tangent, \(\frac{x}{y} = 0\).

Each function supplies valuable insight into the properties and measures of the angle. Understanding these trigonometric ratios is crucial in exploring right triangles and the unit circle.

Radians

Angles can be measured in degrees or radians. Unlike degrees, which are widely used in everyday contexts, radians offer a natural way to consider angles in mathematics. One radian is the angle made by wrapping the radius of a circle around its perimeter.
The concept of radians brings us to the idea that a full circle is \(2\pi\) radians. This equates to \(360^\circ\). Therefore, an angle of \(\frac{3\pi}{2}\) radians corresponds to\(270^\circ\).

• This conversion is crucial in trigonometry and calculus.
• It provides a means to smoothly integrate angles into equations and formulas.

In our problem, the angle's smallest positive measure is \(\frac{3\pi}{2}\) radians because the point (0, -4) lies on a vertical line downward, equivalent to \(270^\circ\).
Through working with radians, you can grasp angles' relationships in more advanced mathematics. It's a vital skill for exploring the broader world of trigonometric and calculus concepts.