To solve for [tex]\( a \)[/tex] in the given equation:
[tex]\[
\frac{a}{\sin 29^\circ} = \frac{16}{\sin 106^\circ}
\][/tex]
you need to isolate [tex]\( a \)[/tex]. Here are the detailed steps:
1. Start with the equation:
[tex]\[
\frac{a}{\sin 29^\circ} = \frac{16}{\sin 106^\circ}
\][/tex]
2. To isolate [tex]\( a \)[/tex], multiply both sides by [tex]\( \sin 29^\circ \)[/tex]:
[tex]\[
a = \left(\frac{16}{\sin 106^\circ}\right) \cdot \sin 29^\circ
\][/tex]
3. To find the numerical value, you need to input the values of [tex]\(\sin 29^\circ\)[/tex] and [tex]\(\sin 106^\circ\)[/tex]:
- [tex]\(\sin 29^\circ \approx 0.4848\)[/tex]
- [tex]\(\sin 106^\circ \approx 0.9613\)[/tex]
4. Substitute these values back into the equation:
[tex]\[
a = \left(\frac{16}{0.9613}\right) \cdot 0.4848
\][/tex]
5. Perform the division inside the brackets first:
[tex]\[
\frac{16}{0.9613} \approx 16.644
\][/tex]
6. Now, multiply the result by [tex]\( \sin 29^\circ \)[/tex]:
[tex]\[
a = 16.644 \times 0.4848 \approx 8.0696
\][/tex]
So, the value of [tex]\( a \)[/tex] is approximately 8.07. Given the options:
a. 7.7
b. 8.1
c. 7.3
d. 12.9
The closest value to 8.07 is [tex]\( \boxed{8.1} \)[/tex].
