Let's solve the problem step by step using the law of sines. We are given:
- Angle [tex]\( A = 51^\circ \)[/tex]
- Side opposite [tex]\( A \)[/tex], [tex]\( a = 2.6 \)[/tex]
- Angle [tex]\( B = 76^\circ \)[/tex]
- Side opposite [tex]\( B \)[/tex], [tex]\( b = z \)[/tex] (to be determined)
- Angle [tex]\( C = 53^\circ \)[/tex]
- Side opposite [tex]\( C \)[/tex], [tex]\( c = 2 \)[/tex]
We will solve these using the law of sines [tex]\(\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}\)[/tex].
### Step 1: Determine the Missing Side [tex]\( z \)[/tex] Opposite Angle [tex]\( B \)[/tex]
The law of sines for angle [tex]\( A \)[/tex] and angle [tex]\( B \)[/tex] gives us:
[tex]\[ \frac{\sin(51^\circ)}{2.6} = \frac{\sin(76^\circ)}{z} \][/tex]
Solving for [tex]\( z \)[/tex]:
[tex]\[ z = \frac{\sin(76^\circ) \cdot 2.6}{\sin(51^\circ)} \][/tex]
Now, the calculated value of [tex]\( z \)[/tex] would be approximately:
[tex]\[ z \approx 3.246 \][/tex]
### Step 2: Verification of Side [tex]\( a = 2.6 \)[/tex]
We can verify [tex]\( a \)[/tex] using another pair of angles and their respective sides. According to the law of sines:
[tex]\[ \frac{\sin(51^\circ)}{2.6} = \frac{\sin(53^\circ)}{2} \][/tex]
Solving for the left-hand side:
[tex]\[ 2.6 \approx \frac{\sin(51^\circ) \cdot 2}{\sin(53^\circ)} \][/tex]
The recalculated [tex]\( a \)[/tex] would be approximately:
[tex]\[ a \approx 1.946 \][/tex]
### Step 3: Verification of Side [tex]\( c = 2 \)[/tex]
We can use sides [tex]\( a \)[/tex] and [tex]\( c \)[/tex] to further confirm our calculations. From the law of sines:
[tex]\[ \frac{\sin(76^\circ)}{2.6} = \frac{\sin(51^\circ)}{z} \][/tex]
Solving for the right-hand side:
[tex]\[ z \approx \frac{\sin(53^\circ) \cdot 2.6}{\sin(51^\circ)} \][/tex]
The recalculated [tex]\( c \)[/tex] would be approximately:
[tex]\[ c \approx 2.672 \][/tex]
### Conclusion:
- The missing side [tex]\( z \)[/tex] opposite to [tex]\( B \)[/tex] is approximately [tex]\( 3.246 \)[/tex].
- Verification of side [tex]\( a \)[/tex] results in approximately [tex]\( 1.946 \)[/tex].
- Verification of side [tex]\( c \)[/tex] results in approximately [tex]\( 2.672 \)[/tex].
Thus, we have determined the missing side [tex]\( z \)[/tex] and verified the sides [tex]\( a \)[/tex] and [tex]\( b \)[/tex] through the given angles and respective sides using the law of sines.
