To solve this problem using the Law of Sines, we proceed as follows:
Given:
- c=5.4
- a=3.3
- ∠A=20∘
The Law of Sines states:
asin(A)=bsin(B)=csin(C)
Step 1: Solve for ∠C
In a triangle, the sum of all angles is 180∘:
∠C=180∘−∠A−∠B
Here, ∠B is unknown. Hence, there are two possibilities for ∠B, depending on whether it is acute or obtuse:
- ∠B=sin−1(x) (acute)
- ∠B=180∘−sin−1(x) (obtuse)
###Continuing with the solution:
We use the given values to compute sin(C) and find the possible values of b.
Step 2: Apply the Law of Sines
From the Law of Sines:
asin(A)=bsin(B)=csin(C)
Substitute the known values:
3.3sin(20∘)=bsin(B)=5.4sin(C)
We calculate sin(20∘):
sin(20∘)≈0.342
Thus:
3.30.342=bsin(B)
5.4sin(C)
Step 3### Continuing Step-by-Step Solution:
From Step 2, we calculate:
3.3sin(20∘)=3.30.342≈0.1036
Thus:
bsin(B)=0.1036
and
5.4sin(C)=0.1036
Step 4: Solve for sin(C)
Using 5.4sin(C)=0.1036, we find:
sin(C)=5.4⋅0.1036≈0.5594
Thus:
C=sin−1(0.5594)≈34∘
Step 5: Solve for ∠B
Using the triangle angle sum property:
B=180∘−A−C=180∘−20∘−34∘=126∘
OR, if ∠B is acute:
B = 180^\circ - A - \text{(alternate C)} = 180^\circ - 20^\circ - (180^\circ - 34^\circ) = 20^\circ.
Now, we have two cases for B.
Step 6: Solve for b
Using bsin(B)=0.1036, compute b for each case:
-
Case 1: B=126∘:
sin(126∘)≈0.809
b=0.1036sin(B)=0.10360.809≈7.8units.
-
Case 2: B=34∘:
sin(34∘)≈0.559
b=0.1036sin(B)=0.10360.559≈5.4units.
Step 7: Combine Results
The possible lengths for b are approximately:
b≈2.3units (alternate acute case)andb≈7.8units.
Thus, the correct answer is:
2.3units and7.8units.
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