Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Discover in-depth answers to your questions from a wide network of experts on our user-friendly Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
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.
- 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.
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.