Absolutely, let's go through this step-by-step:
1. Given Information:
- [tex]\(\sin 2A = \frac{24}{25}\)[/tex]
- [tex]\(\sin A = \frac{4}{5}\)[/tex]
2. Double Angle Identity for Sine:
- We know from trigonometric identities that [tex]\(\sin 2A = 2 \sin A \cos A\)[/tex].
3. Setting Up the Equation:
- Substitute the known values into the double angle identity:
[tex]\[
\sin 2A = 2 \sin A \cos A
\][/tex]
[tex]\[
\frac{24}{25} = 2 \left(\frac{4}{5}\right) \cos A
\][/tex]
4. Solving for [tex]\(\cos A\)[/tex]:
- First, simplify the right-hand side:
[tex]\[
\frac{24}{25} = 2 \cdot \frac{4}{5} \cdot \cos A = \frac{8}{5} \cos A
\][/tex]
- Next, isolate [tex]\(\cos A\)[/tex] by dividing both sides of the equation by [tex]\(\frac{8}{5}\)[/tex]:
[tex]\[
\cos A = \frac{\frac{24}{25}}{\frac{8}{5}}
\][/tex]
- Simplify the division:
[tex]\[
\cos A = \frac{24}{25} \times \frac{5}{8} = \frac{24 \cdot 5}{25 \cdot 8} = \frac{120}{200} = \frac{3}{5}
\][/tex]
5. Final Answer:
- Therefore, the value of [tex]\(\cos A\)[/tex] is:
[tex]\[
\cos A = 0.6
\][/tex]
