Let's find [tex]\(\sin 2A\)[/tex] and [tex]\(\cos 2A\)[/tex] step-by-step given that [tex]\( \sin A = \frac{48}{73} \)[/tex] and [tex]\( \cos A = \frac{55}{73} \)[/tex].
### Step 1: Recall the Double-Angle Formulas:
For any angle [tex]\( A \)[/tex], the double-angle formulas for sine and cosine are:
[tex]\[ \sin 2A = 2 \sin A \cos A \][/tex]
[tex]\[ \cos 2A = \cos^2 A - \sin^2 A \][/tex]
### Step 2: Substitute the Given Values:
We are given:
[tex]\[ \sin A = \frac{48}{73} \][/tex]
[tex]\[ \cos A = \frac{55}{73} \][/tex]
#### Finding [tex]\(\sin 2A\)[/tex]:
Using the formula for [tex]\(\sin 2A\)[/tex]:
[tex]\[ \sin 2A = 2 \sin A \cos A \][/tex]
Substitute the given values:
[tex]\[ \sin 2A = 2 \left(\frac{48}{73}\right) \left(\frac{55}{73}\right) \][/tex]
#### Finding [tex]\(\cos 2A\)[/tex]:
Using the formula for [tex]\(\cos 2A\)[/tex]:
[tex]\[ \cos 2A = \cos^2 A - \sin^2 A \][/tex]
Substitute the given values:
[tex]\[ \cos 2A = \left(\frac{55}{73}\right)^2 - \left(\frac{48}{73}\right)^2 \][/tex]
After substituting the values and following these steps for simplification, we get the values:
[tex]\[ \sin 2A \approx 0.991 \][/tex]
[tex]\[ \cos 2A \approx 0.135 \][/tex]
### Step 3: Write the Final Answer:
[tex]\[
\begin{array}{l}
\sin 2A = 0.9908050290861324 \\
\cos 2A = 0.13529742916119353
\end{array}
\][/tex]
These precise values are the results for [tex]\(\sin 2A\)[/tex] and [tex]\(\cos 2A\)[/tex].
