To solve this problem, we start with the information given: [tex]\(\sin \theta = \frac{0}{85} = 0\)[/tex].
1. Sine ([tex]\(\sin \theta\)[/tex]):
[tex]\[
\sin \theta = 0
\][/tex]
2. Cosine ([tex]\(\cos \theta\)[/tex]):
We use the Pythagorean identity [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex]. Since [tex]\(\sin \theta = 0\)[/tex]:
[tex]\[
0^2 + \cos^2 \theta = 1 \implies \cos^2 \theta = 1 \implies \cos \theta = \sqrt{1} = 1
\][/tex]
In Quadrant I, [tex]\(\cos \theta\)[/tex] is positive, so:
[tex]\[
\cos \theta = 1
\][/tex]
3. Tangent ([tex]\(\tan \theta\)[/tex]):
Tangent is defined as the ratio of sine to cosine:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{0}{1} = 0
\][/tex]
4. Cosecant ([tex]\(\csc \theta\)[/tex]):
Cosecant is the reciprocal of sine:
[tex]\[
\csc \theta = \frac{1}{\sin \theta} = \frac{1}{0}
\][/tex]
Since division by zero is undefined, [tex]\(\csc \theta\)[/tex] is:
[tex]\[
\csc \theta = \infty
\][/tex]
5. Secant ([tex]\(\sec \theta\)[/tex]):
Secant is the reciprocal of cosine:
[tex]\[
\sec \theta = \frac{1}{\cos \theta} = \frac{1}{1} = 1
\][/tex]
6. Cotangent ([tex]\(\cot \theta\)[/tex]):
Cotangent is the reciprocal of tangent:
[tex]\[
\cot \theta = \frac{1}{\tan \theta} = \frac{1}{0}
\][/tex]
Since division by zero is undefined, [tex]\(\cot \theta\)[/tex] is:
[tex]\[
\cot \theta = \infty
\][/tex]
So, the exact values of the remaining trigonometric functions of [tex]\(\theta\)[/tex] are:
[tex]\[
\cos \theta = 1, \quad \tan \theta = 0, \quad \csc \theta = \infty, \quad \sec \theta = 1, \quad \cot \theta = \infty
\][/tex]
