Problem 31

Question

Find the values of the six trigonometric functions of $\boldsymbol{\theta}$ with the given constraint. $$ \begin{aligned} &\cot \theta \text { is undefined. }\\\ &\pi / 2 \leq \theta \leq 3 \pi / 2 \end{aligned} $$

Step-by-Step Solution

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Answer

Given the constraints for $ \theta $, there are two possible sets of values for the trigonometric functions. When $ \theta = \pi/2 $, $\sin \theta = 1$, $\cos \theta = 0$, $\tan \theta$ is undefined, $\csc \theta = 1$, $\sec \theta$ is undefined and $\cot \theta$ is undefined. When $ \theta = 3\pi/2 $, $\sin \theta = -1$, $\cos \theta = 0$, $\tan \theta$ is undefined, $\csc \theta = -1$, $\sec \theta$ is undefined and $\cot \theta$ is undefined.

1Step 1: Find where cotangent is undefined

Since cotangent is defined as the reciprocal of tangent, cotangent is undefined where tangent is zero. On the unit circle, tangent is zero at angles of $ \pi/2 $ and $3\pi/2$. Given the constraint that $ \pi / 2 \leq \theta \leq 3 \pi / 2 $, this means $ \theta $ must be $ \pi/2 $ or $3\pi/2$.

2Step 2: Evaluate other trigonometric functions at $\pi/2$

At $\theta = \pi/2$, the coordinates on the unit circle are (0,1), therefore $\sin \theta = 1$, $\cos \theta = 0$, $\tan \theta $ is undefined, $\csc \theta = 1$, $\sec \theta$ is undefined and $\cot \theta$ is undefined.

3Step 3: Evaluate other trigonometric functions at $3\pi/2$

At $\theta = 3\pi/2$, the coordinates on the unit circle are (0,-1), therefore $\sin \theta = -1$, $\cos \theta = 0$, $\tan \theta $ is undefined, $\csc \theta = -1$, $\sec \theta$ is undefined and $\cot \theta$ is undefined.

Key Concepts

Unit CircleCotangentTangentUndefined Values

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. The circle provides a way to understand the behavior of trigonometric functions for any given angle. The coordinates of points on the unit circle can be represented as

$(\cos \theta, \sin \theta)$

This means for any angle $\theta$, the x-coordinate is the cosine of that angle, and the y-coordinate is the sine.

Understanding the unit circle helps in evaluating trigonometric functions like sine, cosine, and tangent. When the angle is measured counterclockwise from the positive x-axis, it follows the path along the circumference of the unit circle.

Cotangent

The cotangent function, abbreviated as $\cot$, is the reciprocal of the tangent function. It is defined as

$\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}$

This means that cotangent is undefined wherever tangent is zero.

On the unit circle, tangent is zero where the y-coordinate (sine) is zero, leading to undefined points for cotangent. In this exercise, it is given that cotangent is undefined, which specifically occurs at angles like $\pi/2$ and $3\pi/2$ where sine equals zero.

Tangent

Tangent, often denoted as $\tan$, is one of the primary trigonometric functions. It is defined as the ratio of the sine and cosine functions:

$\tan \theta = \frac{\sin \theta}{\cos \theta}$

In the unit circle, tangent functions can be easily visualized. As the y-value changes relative to the x-value, the tangent provides an understanding of the slope of the angle's radius.

Tangent is undefined when the cosine of the angle is zero since division by zero is undefined. This is specifically at the angles $\pi/2$ and $3\pi/2$ in the unit circle.

Undefined Values

In trigonometry, undefined values occur when a function does not yield a finite result. This typically happens when you try to divide by zero or calculate the reciprocal of zero.

For the tangent function, undefined values occur where cosine is zero, making division undefined. For cotangent, undefined values occur where sine is zero.

At $\pi/2$, tangent is undefined because $\cos \theta = 0$.
Similarly, cotangent is undefined at both $\pi/2$ and $3\pi/2$ because $\sin \theta = 0$.

Understanding these points helps in determining when these functions do not provide a numerical output.

Problem 31

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Problem 31

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