Sure, let's find the exact values of the remaining trigonometric functions using the fundamental identities given the following information:
[tex]\[ \cos x = \frac{1}{\sqrt{10}} \][/tex]
[tex]\[ \csc x = \frac{\sqrt{10}}{3} \][/tex]
We will derive each of the remaining trigonometric functions step by step.
### Step 1: Find [tex]\(\sin x\)[/tex]
We know from the reciprocal identity that:
[tex]\[ \sin x = \frac{1}{\csc x} \][/tex]
Substituting the given value of [tex]\(\csc x\)[/tex]:
[tex]\[ \sin x = \frac{1}{\frac{\sqrt{10}}{3}} \][/tex]
[tex]\[ \sin x = \frac{3}{\sqrt{10}} \][/tex]
To simplify:
[tex]\[ \sin x = 0.9486832980505138 \][/tex]
### Step 2: Find [tex]\(\tan x\)[/tex]
We use the quotient identity:
[tex]\[ \tan x = \frac{\sin x}{\cos x} \][/tex]
Substituting the known values:
[tex]\[ \tan x = \frac{0.9486832980505138}{0.31622776601683794} \][/tex]
[tex]\[ \tan x = 3.0 \][/tex]
### Step 3: Find [tex]\(\sec x\)[/tex]
We know from the reciprocal identity that:
[tex]\[ \sec x = \frac{1}{\cos x} \][/tex]
Substituting the given value of [tex]\(\cos x\)[/tex]:
[tex]\[ \sec x = \frac{1}{0.31622776601683794} \][/tex]
[tex]\[ \sec x = 3.162277660168379 \][/tex]
### Step 4: Find [tex]\(\cot x\)[/tex]
We know from the reciprocal identity that:
[tex]\[ \cot x = \frac{1}{\tan x} \][/tex]
Substituting the known value of [tex]\(\tan x\)[/tex]:
[tex]\[ \cot x = \frac{1}{3.0} \][/tex]
[tex]\[ \cot x = 0.3333333333333333 \][/tex]
Now we have all the trigonometric functions:
[tex]\[
\begin{array}{l}
\sin x = 0.9486832980505138 \\
\cos x = 0.31622776601683794 \\
\tan x = 3.0 \\
\csc x = 1.0540925533894598 \\
\sec x = 3.162277660168379 \\
\cot x = 0.3333333333333333
\end{array}
\][/tex]
