To solve for the trigonometric ratios given that [tex]\(\cot \theta = \frac{20}{21}\)[/tex], we start by considering the relationships between the trigonometric functions.
1. Find [tex]\(\tan \theta\)[/tex]
We know that:
[tex]\[
\cot \theta = \frac{1}{\tan \theta}
\][/tex]
Given [tex]\(\cot \theta = \frac{20}{21}\)[/tex], we can find [tex]\(\tan \theta\)[/tex] by taking the reciprocal:
[tex]\[
\tan \theta = \frac{1}{\cot \theta} = \frac{21}{20}
\][/tex]
2. Setup the right triangle
The tangent of [tex]\(\theta\)[/tex] is defined as the ratio of the opposite side to the adjacent side:
[tex]\[
\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{21}{20}
\][/tex]
Assume we are dealing with a right triangle where the side opposite [tex]\(\theta\)[/tex] is 21 units and the side adjacent to [tex]\(\theta\)[/tex] is 20 units.
3. Calculate the hypotenuse using the Pythagorean theorem
The hypotenuse, [tex]\(h\)[/tex], can be found using:
[tex]\[
h = \sqrt{(\text{opposite})^2 + (\text{adjacent})^2} = \sqrt{21^2 + 20^2}
\][/tex]
Calculate:
[tex]\[
h = \sqrt{441 + 400} = \sqrt{841} = 29
\][/tex]
4. Find [tex]\(\sin \theta\)[/tex]
The sine of [tex]\(\theta\)[/tex] is the ratio of the opposite side to the hypotenuse:
[tex]\[
\sin \theta = \frac{\text{opposite}}{h} = \frac{21}{29}
\][/tex]
5. Find [tex]\(\cos \theta\)[/tex]
The cosine of [tex]\(\theta\)[/tex] is the ratio of the adjacent side to the hypotenuse:
[tex]\[
\cos \theta = \frac{\text{adjacent}}{h} = \frac{20}{29}
\][/tex]
Based on these calculations, the trigonometric ratios are as follows:
- [tex]\(\sin \theta \approx 0.7241 \)[/tex]
- [tex]\(\cos \theta \approx 0.6897 \)[/tex]
- [tex]\(\tan \theta = 1.05 \)[/tex]
Therefore:
[tex]\[
\sin \theta = 0.7241379310344828, \quad \cos \theta = 0.6896551724137931, \quad \tan \theta = 1.05
\][/tex]
