Certainly! Let's solve this step by step.
Given:
[tex]\[
\sin \theta = \frac{84}{85}
\][/tex]
### Step 1: Calculate [tex]\(\cos \theta\)[/tex]
Using the Pythagorean identity:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
Plug in the value of [tex]\(\sin \theta\)[/tex]:
[tex]\[
\left( \frac{84}{85} \right)^2 + \cos^2 \theta = 1
\][/tex]
Calculate [tex]\(\left( \frac{84}{85} \right)^2\)[/tex]:
[tex]\[
\left( \frac{84}{85} \right)^2 = \frac{7056}{7225}
\][/tex]
So the equation becomes:
[tex]\[
\frac{7056}{7225} + \cos^2 \theta = 1
\][/tex]
Subtract [tex]\(\frac{7056}{7225}\)[/tex] from both sides:
[tex]\[
\cos^2 \theta = 1 - \frac{7056}{7225} = \frac{7225}{7225} - \frac{7056}{7225} = \frac{169}{7225}
\][/tex]
Taking the square root of both sides:
[tex]\[
\cos \theta = \sqrt{\frac{169}{7225}} = \frac{\sqrt{169}}{\sqrt{7225}} = \frac{13}{85}
\][/tex]
Since [tex]\(\theta\)[/tex] is in Quadrant I, [tex]\(\cos \theta\)[/tex] is positive:
[tex]\[
\cos \theta = \frac{13}{85}
\][/tex]
### Step 2: Calculate [tex]\(\tan \theta\)[/tex]
Using the definition of tangent:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
Substitute the given and calculated values:
[tex]\[
\tan \theta = \frac{84}{85} \div \frac{13}{85} = \frac{84}{13}
\][/tex]
Simplifying the fraction:
[tex]\[
\tan \theta = \frac{84}{13} = 6.4615384615384615
\][/tex]
### Step 3: Calculate [tex]\(\csc \theta\)[/tex]
Using the definition of cosecant:
[tex]\[
\csc \theta = \frac{1}{\sin \theta}
\][/tex]
Substitute the given value:
[tex]\[
\csc \theta = \frac{1}{\frac{84}{85}} = \frac{85}{84} = 1.0119047619047619
\][/tex]
### Step 4: Calculate [tex]\(\sec \theta\)[/tex]
Using the definition of secant:
[tex]\[
\sec \theta = \frac{1}{\cos \theta}
\][/tex]
Substitute the calculated value:
[tex]\[
\sec \theta = \frac{1}{\frac{13}{85}} = \frac{85}{13} = 6.538461538461538
\][/tex]
### Step 5: Calculate [tex]\(\cot \theta\)[/tex]
Using the definition of cotangent:
[tex]\[
\cot \theta = \frac{1}{\tan \theta}
\][/tex]
Substitute the calculated value:
[tex]\[
\cot \theta = \frac{1}{6.4615384615384615} = 0.15476190476190448
\][/tex]
### Final Answer:
The six trigonometric functions of [tex]\(\theta\)[/tex] are:
[tex]\[
\sin \theta = \frac{84}{85}, \quad \cos \theta = \frac{13}{85}, \quad \tan \theta = 6.4615384615384615,
\][/tex]
[tex]\[
\csc \theta = 1.0119047619047619, \quad \sec \theta = 6.538461538461538, \quad \cot \theta = 0.15476190476190448
\][/tex]
