To find [tex]\(\cos (2A)\)[/tex] given that [tex]\(\sin(A) = \frac{65}{97}\)[/tex] and knowing that [tex]\(A\)[/tex] is in the first quadrant, follow these steps:
1. Determine [tex]\(\cos(A)\)[/tex]:
Since [tex]\(A\)[/tex] is in the first quadrant, [tex]\(\cos(A)\)[/tex] will be positive. We use the Pythagorean identity:
[tex]\[
\sin^2(A) + \cos^2(A) = 1
\][/tex]
Given [tex]\(\sin(A) = \frac{65}{97}\)[/tex], we first find [tex]\(\sin^2(A)\)[/tex]:
[tex]\[
\sin^2(A) = \left(\frac{65}{97}\right)^2 = \frac{4225}{9409}
\][/tex]
Using the Pythagorean identity, [tex]\(\cos^2(A)\)[/tex] is calculated as:
[tex]\[
\cos^2(A) = 1 - \sin^2(A) = 1 - \frac{4225}{9409} = \frac{9409}{9409} - \frac{4225}{9409} = \frac{5184}{9409}
\][/tex]
Thus, [tex]\(\cos(A)\)[/tex] is:
[tex]\[
\cos(A) = \sqrt{\frac{5184}{9409}} = \frac{\sqrt{5184}}{\sqrt{9409}} = \frac{72}{97}
\][/tex]
2. Determine [tex]\(\cos(2A)\)[/tex]:
Using the double-angle formula for cosine, [tex]\(\cos(2A)\)[/tex] is given by:
[tex]\[
\cos(2A) = \cos^2(A) - \sin^2(A)
\][/tex]
We have already found [tex]\(\cos^2(A)\)[/tex] and [tex]\(\sin^2(A)\)[/tex] as:
[tex]\[
\cos^2(A) = \frac{5184}{9409}
\][/tex]
[tex]\[
\sin^2(A) = \frac{4225}{9409}
\][/tex]
Substituting these values in, we get:
[tex]\[
\cos(2A) = \frac{5184}{9409} - \frac{4225}{9409} = \frac{5184 - 4225}{9409} = \frac{959}{9409}
\][/tex]
Therefore, the exact values obtained are:
[tex]\[
\cos(A) = \frac{72}{97} \approx 0.7423
\][/tex]
[tex]\[
\cos(2A) = \frac{959}{9409} \approx 0.1019
\][/tex]
