Chapter 4.2, Problem 54E

(a)

To determine

Find the value of $\cot (-\text{t})\text{if}\cot \text{t}=-\frac{1}{4}$ by using the Trigonometric Rule.

Answer to Problem 54E

  $\frac{1}{4}$

Explanation of Solution

Given: The angle $\cot \text{t}=-\frac{1}{4}$

Concept Used:

In Trigonometry we know that: $\text{sin (θ}\text{+}\text{2πn}\text{)}\text{=}\pm \text{sin}\text{θ}\text{;}\text{cos}\text{(θ}\text{+}\text{2πn}\text{)}=\pm \cos \text{θ}$ (positive or negative) depending on the values of n and the quadrant

For any integer $\text{n}$ and real number $\text{t}=\text{θ}$ , the functions that behave in such a repetitive (cyclic) manner are called periodic.

Sine and Cosine functions are periodic and have a period $\text{2π}$

The cosine and secant functions are even

  $\text{cos}(-\text{θ})=\cos \text{θ}\text{;}\sec (-\text{θ})=\sec \text{θ}$

The sine, cosecant, tangent and cotangent functions are odd.

  $\begin{array}{l}\sin (-\text{θ})=-\sin \text{θ}\text{;}\csc (-\text{θ})=-\csc \text{θ}\text{;}\\ \text{tan}(-\text{θ})=-\tan \text{θ}\text{;}\text{cot}(-\text{θ})=-\cot \text{θ}\text{;}\end{array}$

Calculation:

Using the rule: $\text{tan}(-\text{θ})=-\tan \text{θ}\text{;}\text{cot}(-\text{θ})=-\cot \text{θ}\text{;}$

  $\text{cot}(-\text{t})=-\cot \text{t}\text{;}$

Given: $\cot \text{t}=-\frac{1}{4}$

  $\begin{array}{l}\text{cot}(-\text{t})=-\cot \text{t}=-(-\frac{1}{4})=\frac{1}{4}\\ \text{cot}(-\text{t})=\frac{1}{4}\end{array}$

Thus the value of $\text{cot}(-\text{t})=\frac{1}{4}$

(b)

To determine

Find the value of $\tan (-\text{t})\text{if}\cot \text{t}=-\frac{1}{4}$ by using the Trigonometric Rule

Answer to Problem 54E

  $-\frac{5}{4}$

Explanation of Solution

Given: The angle $\cot \text{t}=-\frac{1}{4}$

Concept Used:

In Trigonometry we know that: $\text{sin (θ}\text{+}\text{2πn}\text{)}\text{=}\pm \text{sin}\text{θ}\text{;}\text{cos}\text{(θ}\text{+}\text{2πn}\text{)}=\pm \cos \text{θ}$ (positive or negative) depending on the values of n and the quadrant

For any integer $\text{n}$ and real number $\text{t}=\text{θ}$ , the functions that behave in such a repetitive (cyclic) manner are called periodic.

Sine and Cosine functions are periodic and have a period $\text{2π}$

The cosine and secant functions are even

  $\text{cos}(-\text{θ})=\cos \text{θ}\text{;}\sec (-\text{θ})=\sec \text{θ}$

The sine, cosecant, tangent and cotangent functions are odd.

  $\begin{array}{l}\sin (-\text{θ})=-\sin \text{θ}\text{;}\csc (-\text{θ})=-\csc \text{θ}\text{;}\\ \text{tan}(-\text{θ})=-\tan \text{θ}\text{;}\text{cot}(-\text{θ})=-\cot \text{θ}\text{;}\end{array}$

Calculation:

Using the rule: $\cot \text{t}=\frac{1}{\tan \text{t}}$

Given: $\cot \text{t}=-\frac{1}{4}$

  $\begin{array}{l}\text{cot}(-\text{t})=\frac{1}{4}\\ \text{tan}(-\text{t})=-\tan \text{t}=-\frac{1}{\cot \text{t}}=-\frac{1}{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}=4\\ \text{tan}(-\text{t})=4\end{array}$

Thus the value of $\text{tan}(-\text{t})=4$