Sure, let's solve this step-by-step using the sine rule. The sine rule states:
[tex]\[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} \][/tex]
Given values:
- [tex]\( a = 9 \)[/tex]
- [tex]\( \angle A = 42^\circ \)[/tex]
- [tex]\( \angle B = 63^\circ \)[/tex]
We need to find the value in the box, which corresponds to [tex]\( b \)[/tex]. Let’s denote the unknown side as [tex]\( b \)[/tex].
First, write down the sine rule equation:
[tex]\[ \frac{9}{\sin(42^\circ)} = \frac{b}{\sin(63^\circ)} \][/tex]
Now we need to find the sine of the angles [tex]\( 42^\circ \)[/tex] and [tex]\( 63^\circ \)[/tex]. Using trigonometric tables or a calculator:
[tex]\[ \sin(42^\circ) \approx 0.6691306063588582 \][/tex]
[tex]\[ \sin(63^\circ) \approx 0.8910065241883678 \][/tex]
Now substitute these values into the equation:
[tex]\[ \frac{9}{0.6691306063588582} = \frac{b}{0.8910065241883678} \][/tex]
Next, isolate [tex]\( b \)[/tex] on one side by multiplying both sides of the equation by [tex]\( \sin(63^\circ) \)[/tex]:
[tex]\[ b = \frac{9 \cdot 0.8910065241883678}{0.6691306063588582} \][/tex]
Calculating this:
[tex]\[ b \approx \frac{8.01905871769531}{0.6691306063588582} \approx 11.984295205583 \][/tex]
So, the value in the box to complete the equation is:
[tex]\[ b \approx 11.984295205583 \][/tex]
