Questions: Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. (If an answer does not exist, enter DNE. Round your answers to one decimal place. Below, enter your answers so that angle B1 is larger than angle B2.) a=32, c=42, angle A = 33° angle B1 = square°, angle B2 = square° angle C1 = C2 = C2 = b° b1 = b2

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. (If an answer does not exist, enter DNE. Round your answers to one decimal place. Below, enter your answers so that angle B1 is larger than angle B2.)

a=32, c=42, angle A = 33°

angle B1 = square°, angle B2 = square°

angle C1 = C2 = C2 = b°

b1 = b2
Transcript text: Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. (If an answer does not exist, enter DNE. Round your answers to one decimal place. Below, enter your answers so that $\angle B_{1}$ is larger than $\angle B_{2}$.) \[ \begin{array}{rlrl} & a=32, \quad c=42, \quad \angle A & =33^{\circ} \\ \angle B_{1} & =\square^{\circ}, \quad \angle B_{2} & =\square^{\circ} \\ \angle C_{1} & =C_{2} & =C_{2} & =b^{\circ} \\ b_{1} & =b_{2} \end{array} \]
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Solution

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Solution Steps

To solve the triangle using the Law of Sines, follow these steps:

  1. Use the Law of Sines: Start by using the Law of Sines to find angle C C . The formula is asinA=csinC\frac{a}{\sin A} = \frac{c}{\sin C}.

  2. Calculate Angle C C : Rearrange the formula to solve for sinC\sin C and then find C C using the inverse sine function.

  3. 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 .

  4. Find Angle B B : Use the fact that the sum of angles in a triangle is 180 180^\circ to find angle B B for each possible triangle.

  5. Calculate Side b b : Use the Law of Sines again to find side b b for each triangle.

Step 1: Calculate sinC \sin C

Using the Law of Sines, we have:

asinA=csinC \frac{a}{\sin A} = \frac{c}{\sin C}

Substituting the known values:

32sin(33)=42sinC \frac{32}{\sin(33^\circ)} = \frac{42}{\sin C}

Calculating sinC \sin C :

sinC=42sin(33)320.7148 \sin C = \frac{42 \cdot \sin(33^\circ)}{32} \approx 0.7148

Step 2: Determine Possible Angles for C C

Since sinC0.7148 \sin C \approx 0.7148 , we can find the angles C1 C_1 and C2 C_2 :

C1=arcsin(0.7148)45.4 C_1 = \arcsin(0.7148) \approx 45.4^\circ C2=180C1134.6 C_2 = 180^\circ - C_1 \approx 134.6^\circ

Step 3: Calculate Corresponding Angles B B

Using the triangle angle sum property A+B+C=180 A + B + C = 180^\circ :

For C1 C_1 :

B1=1803345.4101.6 B_1 = 180^\circ - 33^\circ - 45.4^\circ \approx 101.6^\circ

For C2 C_2 :

B2=18033134.612.4 B_2 = 180^\circ - 33^\circ - 134.6^\circ \approx 12.4^\circ

Step 4: Calculate Side b b

Using the Law of Sines again to find side b b :

For B1 B_1 :

b1=asin(B1)sin(A)=32sin(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

For B2 B_2 :

b2=asin(B2)sin(A)=32sin(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

Final Answer

The angles and corresponding side lengths are:

B1101.6,B212.4 B_1 \approx 101.6^\circ, \quad B_2 \approx 12.4^\circ C145.4,C2134.6 C_1 \approx 45.4^\circ, \quad C_2 \approx 134.6^\circ b156.5,b214.5 b_1 \approx 56.5, \quad b_2 \approx 14.5

Thus, the final boxed answers are:

B1101.6,B212.4 \boxed{B_1 \approx 101.6^\circ, \quad B_2 \approx 12.4^\circ}

C145.4,C2134.6 \boxed{C_1 \approx 45.4^\circ, \quad C_2 \approx 134.6^\circ}

b156.5,b214.5 \boxed{b_1 \approx 56.5, \quad b_2 \approx 14.5}

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