To find [tex]\(\cos(t)\)[/tex] given that [tex]\(\tan(t) = \frac{5}{12}\)[/tex] and [tex]\(0 < t < \frac{\pi}{2}\)[/tex], we can use trigonometric identities involving the tangent and secant functions. Here's a step-by-step process:
### Step 1: Use the trigonometric identity involving tangent
We know that:
[tex]\[ \tan^2(t) + 1 = \sec^2(t) \][/tex]
### Step 2: Substitute the given value of [tex]\(\tan(t)\)[/tex]
Given [tex]\(\tan(t) = \frac{5}{12}\)[/tex], we first calculate [tex]\(\tan^2(t)\)[/tex]:
[tex]\[ \tan^2(t) = \left(\frac{5}{12}\right)^2 = \frac{25}{144} \][/tex]
So, we substitute [tex]\(\tan^2(t)\)[/tex] into the identity:
[tex]\[ \sec^2(t) = 1 + \frac{25}{144} \][/tex]
### Step 3: Simplify to find [tex]\(\sec^2(t)\)[/tex]
Calculate the sum:
[tex]\[ \sec^2(t) = 1 + \frac{25}{144} \][/tex]
To add these, first express 1 as a fraction with a denominator of 144:
[tex]\[ 1 = \frac{144}{144} \][/tex]
Now add the two fractions:
[tex]\[ \sec^2(t) = \frac{144}{144} + \frac{25}{144} = \frac{169}{144} \][/tex]
So:
[tex]\[ \sec^2(t) = 1.1736111111111112 \][/tex]
### Step 4: Calculate [tex]\(\sec(t)\)[/tex]
Take the square root of [tex]\(\sec^2(t)\)[/tex] to find [tex]\(\sec(t)\)[/tex]:
[tex]\[ \sec(t) = \sqrt{1.1736111111111112} \][/tex]
### Step 5: Find [tex]\(\cos(t)\)[/tex]
We know that [tex]\(\sec(t)\)[/tex] is the reciprocal of [tex]\(\cos(t)\)[/tex]:
[tex]\[ \sec(t) = \frac{1}{\cos(t)} \][/tex]
Thus, [tex]\(\cos(t)\)[/tex] is the reciprocal of [tex]\(\sec(t)\)[/tex]:
[tex]\[ \cos(t) = \frac{1}{\sqrt{1.1736111111111112}} \][/tex]
[tex]\[ \cos(t) = 0.9230769230769231 \][/tex]
In summary, given that [tex]\(\tan(t) = \frac{5}{12}\)[/tex], we've calculated [tex]\(\sec^2(t)\)[/tex] to be 1.1736111111111112, and then found that [tex]\(\cos(t) = 0.9230769230769231\)[/tex].
