To solve the triangle using the Law of Sines, follow these steps:
Use the Law of Sines: Start by using the Law of Sines to find angle C C C. The formula is asinA=csinC\frac{a}{\sin A} = \frac{c}{\sin C}sinAa=sinCc.
Calculate Angle C C C: Rearrange the formula to solve for sinC\sin CsinC and then find C C C using the inverse sine function.
Determine Possible Triangles: Since the sine function can have two possible angles (an acute and an obtuse angle), check for both possibilities of angle C C C.
Find Angle B B B: Use the fact that the sum of angles in a triangle is 180∘ 180^\circ 180∘ to find angle B B B for each possible triangle.
Calculate Side b b b: Use the Law of Sines again to find side b b b for each triangle.
Using the Law of Sines, we have:
asinA=csinC \frac{a}{\sin A} = \frac{c}{\sin C} sinAa=sinCc
Substituting the known values:
32sin(33∘)=42sinC \frac{32}{\sin(33^\circ)} = \frac{42}{\sin C} sin(33∘)32=sinC42
Calculating sinC \sin C sinC:
sinC=42⋅sin(33∘)32≈0.7148 \sin C = \frac{42 \cdot \sin(33^\circ)}{32} \approx 0.7148 sinC=3242⋅sin(33∘)≈0.7148
Since sinC≈0.7148 \sin C \approx 0.7148 sinC≈0.7148, we can find the angles C1 C_1 C1 and C2 C_2 C2:
C1=arcsin(0.7148)≈45.4∘ C_1 = \arcsin(0.7148) \approx 45.4^\circ C1=arcsin(0.7148)≈45.4∘ C2=180∘−C1≈134.6∘ C_2 = 180^\circ - C_1 \approx 134.6^\circ C2=180∘−C1≈134.6∘
Using the triangle angle sum property A+B+C=180∘ A + B + C = 180^\circ A+B+C=180∘:
For C1 C_1 C1:
B1=180∘−33∘−45.4∘≈101.6∘ B_1 = 180^\circ - 33^\circ - 45.4^\circ \approx 101.6^\circ B1=180∘−33∘−45.4∘≈101.6∘
For C2 C_2 C2:
B2=180∘−33∘−134.6∘≈12.4∘ B_2 = 180^\circ - 33^\circ - 134.6^\circ \approx 12.4^\circ B2=180∘−33∘−134.6∘≈12.4∘
Using the Law of Sines again to find side b b b:
For B1 B_1 B1:
b1=a⋅sin(B1)sin(A)=32⋅sin(101.6∘)sin(33∘)≈56.5 b_1 = \frac{a \cdot \sin(B_1)}{\sin(A)} = \frac{32 \cdot \sin(101.6^\circ)}{\sin(33^\circ)} \approx 56.5 b1=sin(A)a⋅sin(B1)=sin(33∘)32⋅sin(101.6∘)≈56.5
For B2 B_2 B2:
b2=a⋅sin(B2)sin(A)=32⋅sin(12.4∘)sin(33∘)≈14.5 b_2 = \frac{a \cdot \sin(B_2)}{\sin(A)} = \frac{32 \cdot \sin(12.4^\circ)}{\sin(33^\circ)} \approx 14.5 b2=sin(A)a⋅sin(B2)=sin(33∘)32⋅sin(12.4∘)≈14.5
The angles and corresponding side lengths are:
B1≈101.6∘,B2≈12.4∘ B_1 \approx 101.6^\circ, \quad B_2 \approx 12.4^\circ B1≈101.6∘,B2≈12.4∘ C1≈45.4∘,C2≈134.6∘ C_1 \approx 45.4^\circ, \quad C_2 \approx 134.6^\circ C1≈45.4∘,C2≈134.6∘ b1≈56.5,b2≈14.5 b_1 \approx 56.5, \quad b_2 \approx 14.5 b1≈56.5,b2≈14.5
Thus, the final boxed answers are:
B1≈101.6∘,B2≈12.4∘ \boxed{B_1 \approx 101.6^\circ, \quad B_2 \approx 12.4^\circ} B1≈101.6∘,B2≈12.4∘
C1≈45.4∘,C2≈134.6∘ \boxed{C_1 \approx 45.4^\circ, \quad C_2 \approx 134.6^\circ} C1≈45.4∘,C2≈134.6∘
b1≈56.5,b2≈14.5 \boxed{b_1 \approx 56.5, \quad b_2 \approx 14.5} b1≈56.5,b2≈14.5
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