To determine which of these given relationships are true, we need to utilize our knowledge of trigonometric identities.
Given:
- [tex]\(\cot U = 0.52\)[/tex]
- [tex]\(\csc U = 1.13\)[/tex]
- [tex]\(\sec T = 0.47\)[/tex]
- [tex]\(\sin T = 0.47\)[/tex]
Let's recall the trigonometric definitions and their relationships:
1. [tex]\(\cot U\)[/tex] is the reciprocal of [tex]\(\tan U\)[/tex]:
[tex]\[
\cot U = \frac{1}{\tan U}
\][/tex]
2. [tex]\(\csc U\)[/tex] is the reciprocal of [tex]\(\sin U\)[/tex]:
[tex]\[
\csc U = \frac{1}{\sin U}
\][/tex]
3. [tex]\(\sec T\)[/tex] is the reciprocal of [tex]\(\cos T\)[/tex]:
[tex]\[
\sec T = \frac{1}{\cos T}
\][/tex]
4. [tex]\(\sin T\)[/tex] is the value itself given directly.
Using the given values:
### Analyzing [tex]\(\cot U = 0.52\)[/tex]
Using the identity for [tex]\(\cot U\)[/tex]:
[tex]\[
\cot U = 0.52 \implies \tan U = \frac{1}{\cot U} = \frac{1}{0.52} \approx 1.923
\][/tex]
This relationship is mathematically consistent and does not present any conflict. Thus, [tex]\(\cot U = 0.52\)[/tex] is a true relationship.
### Analyzing [tex]\(\csc U = 1.13\)[/tex]
Using the identity for [tex]\(\csc U\)[/tex]:
[tex]\[
\csc U = 1.13 \implies \sin U = \frac{1}{\csc U} = \frac{1}{1.13} \approx 0.885
\][/tex]
This also seems consistent. Thus, [tex]\(\csc U = 1.13\)[/tex] is a true relationship.
### Analyzing [tex]\(\sec T = 0.47\)[/tex]
Using the identity for [tex]\(\sec T\)[/tex]:
[tex]\[
\sec T = 0.47 \implies \cos T = \frac{1}{\sec T} = \frac{1}{0.47} \approx 2.128
\][/tex]
However, the value of [tex]\(\cos T\)[/tex] must lie between -1 and 1 inclusive, as the cosine function cannot have a value outside this range. Therefore, [tex]\(\cos T \approx 2.128\)[/tex] is not possible. Thus, [tex]\(\sec T = 0.47\)[/tex] is not a true relationship.
### Analyzing [tex]\(\sin T = 0.47\)[/tex]
This value is directly given. Since [tex]\(\sin T\)[/tex] can indeed be any value between -1 and 1, including 0.47, this relationship is valid.
Thus, the true relationships that hold in this scenario are:
- [tex]\(\cot U = 0.52\)[/tex]
- [tex]\(\csc U = 1.13\)[/tex]
- [tex]\(\sin T = 0.47\)[/tex]
Therefore, the correct answers are:
- a. [tex]\(\cot U=0.52\)[/tex]
- b. [tex]\(\csc U=1.13\)[/tex]
- d. [tex]\(\sin T=0.47\)[/tex]
