Math / Geometry

QuestionSolve the triangle, if possible. $a=23.5 \mathrm{~cm}, b=11.77 \mathrm{~cm}, A=32.1^{\circ}$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. (Round degree measures to the nearest tenth as needed. Round side measures to the nearest hundredth as needed.) A. There are 2 possible solutions to the triangle. The measurements for the solution with the longer side $c$ are as follows. $B \approx$ $\square$ $C \approx$ 0 $c \approx$ $\square$ cm The measurements for the solution with the shorter side c are as follows. $\wedge=$ $\square$ $\mathrm{C} \approx$ $\square$ $c \approx$ $\square$ cm B. There is 1 possible solution to the triangle. The measurements for the remaining angles $B$ and $C$ and side $C$ are as follows. $B \approx$ $\square$ $\mathrm{C} \approx$ $\square$ $c \approx$ $\square$ cm C. There are no possible solutions for the triangle.

Studdy Solution

STEP 1

What is this asking? We're given two sides ( $a$ and $b$ ) and an angle ( $A$ ) of a triangle, and we need to find the other side ( $c$ ) and the other two angles ( $B$ and $C$ ).
We also need to figure out if there are multiple possible triangles or no triangle at all! Watch out! Remember the ambiguous case of the sine rule!
Sometimes, with this setup, you can have two different triangles that fit the given information, or maybe even none at all!

STEP 2

1. Use the sine rule to find angle $B$ .
2. Check for the ambiguous case.
3. Find angle $C$ .
4. Use the sine rule to find side $c$ .

STEP 3

The sine rule says that $\frac{\sin(A)}{a} = \frac{\sin(B)}{b}$ .
We know $a = \textbf{23.5 cm}$ , $b = \textbf{11.77 cm}$ , and $A = \textbf{32.1}^{\circ}$ .
Let's plug those values in!

STEP 4

So, we have $\frac{\sin(32.1^{\circ})}{23.5} = \frac{\sin(B)}{11.77}$ .

STEP 5

To get $\sin(B)$ by itself, we can multiply both sides of the equation by 11.77: $\sin(B) = \frac{11.77 \cdot \sin(32.1^{\circ})}{23.5}$

STEP 6

Calculating the right side gives us: $\sin(B) \approx 0.2621$

STEP 7

Now, take the inverse sine (arcsin) of both sides to find $B$ : $B = \arcsin(0.2621)$ $B \approx \textbf{15.2}^{\circ}$

STEP 8

Since $\sin(B) = \sin(180^{\circ} - B)$ , there might be another possible angle $B$ .
Let's calculate the supplementary angle: $180^{\circ} - 15.2^{\circ} = \textbf{164.8}^{\circ}$ .

STEP 9

If $B = 164.8^{\circ}$ , then $A + B = 32.1^{\circ} + 164.8^{\circ} = 196.9^{\circ}$ .
This is greater than $180^{\circ}$ , which is impossible for a triangle!
So, there's only one possible solution for $B$ .

STEP 10

The angles in a triangle add up to $180^{\circ}$ .
We know $A = 32.1^{\circ}$ and $B \approx 15.2^{\circ}$ , so $C = 180^{\circ} - A - B$ .

STEP 11

$C = 180^{\circ} - 32.1^{\circ} - 15.2^{\circ}$ $C \approx \textbf{132.7}^{\circ}$

STEP 12

We can use the sine rule again: $\frac{\sin(A)}{a} = \frac{\sin(C)}{c}$ .

STEP 13

We have $\frac{\sin(32.1^{\circ})}{23.5} = \frac{\sin(132.7^{\circ})}{c}$ .

STEP 14

Multiply both sides by $c$ and $23.5$ , then divide by $\sin(32.1^{\circ})$ : $c = \frac{23.5 \cdot \sin(132.7^{\circ})}{\sin(32.1^{\circ})}$

STEP 15

$c \approx \textbf{33.69} \text{ cm}$

STEP 16

There is one possible solution to the triangle.
The measurements for the remaining angles $B$ and $C$ and side $c$ are as follows: $B \approx 15.2^{\circ}$ , $C \approx 132.7^{\circ}$ , and $c \approx 33.69$ cm.

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Studdy Solution

STEP 1

STEP 2

1. Use the sine rule to find angle $B$ .
2. Check for the ambiguous case.
3. Find angle $C$ .
4. Use the sine rule to find side $c$ .

STEP 3

The sine rule says that $\frac{\sin(A)}{a} = \frac{\sin(B)}{b}$ .
We know $a = \textbf{23.5 cm}$ , $b = \textbf{11.77 cm}$ , and $A = \textbf{32.1}^{\circ}$ .
Let's plug those values in!

STEP 4

So, we have $\frac{\sin(32.1^{\circ})}{23.5} = \frac{\sin(B)}{11.77}$ .

STEP 5

To get $\sin(B)$ by itself, we can multiply both sides of the equation by 11.77: $\sin(B) = \frac{11.77 \cdot \sin(32.1^{\circ})}{23.5}$

STEP 6

Calculating the right side gives us: $\sin(B) \approx 0.2621$

STEP 7

Now, take the inverse sine (arcsin) of both sides to find $B$ : $B = \arcsin(0.2621)$ $B \approx \textbf{15.2}^{\circ}$

STEP 8

Since $\sin(B) = \sin(180^{\circ} - B)$ , there might be another possible angle $B$ .
Let's calculate the supplementary angle: $180^{\circ} - 15.2^{\circ} = \textbf{164.8}^{\circ}$ .

STEP 9

If $B = 164.8^{\circ}$ , then $A + B = 32.1^{\circ} + 164.8^{\circ} = 196.9^{\circ}$ .
This is greater than $180^{\circ}$ , which is impossible for a triangle!
So, there's only one possible solution for $B$ .

STEP 10

The angles in a triangle add up to $180^{\circ}$ .
We know $A = 32.1^{\circ}$ and $B \approx 15.2^{\circ}$ , so $C = 180^{\circ} - A - B$ .

STEP 11

$C = 180^{\circ} - 32.1^{\circ} - 15.2^{\circ}$ $C \approx \textbf{132.7}^{\circ}$

STEP 12

We can use the sine rule again: $\frac{\sin(A)}{a} = \frac{\sin(C)}{c}$ .

STEP 13

We have $\frac{\sin(32.1^{\circ})}{23.5} = \frac{\sin(132.7^{\circ})}{c}$ .

STEP 14

Multiply both sides by $c$ and $23.5$ , then divide by $\sin(32.1^{\circ})$ : $c = \frac{23.5 \cdot \sin(132.7^{\circ})}{\sin(32.1^{\circ})}$

STEP 15

$c \approx \textbf{33.69} \text{ cm}$

STEP 16

There is one possible solution to the triangle.
The measurements for the remaining angles $B$ and $C$ and side $c$ are as follows: $B \approx 15.2^{\circ}$ , $C \approx 132.7^{\circ}$ , and $c \approx 33.69$ cm.

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Studdy Solution

STEP 1

STEP 2

1. Use the sine rule to find angle BBB.2. Check for the ambiguous case.3. Find angle CCC.4. Use the sine rule to find side ccc.

STEP 3

The sine rule says that sin⁡(A)a=sin⁡(B)b\frac{\sin(A)}{a} = \frac{\sin(B)}{b}asin(A)​=bsin(B)​.We know a=23.5 cma = \textbf{23.5 cm}a=23.5 cm, b=11.77 cmb = \textbf{11.77 cm}b=11.77 cm, and A=32.1∘A = \textbf{32.1}^{\circ}A=32.1∘.Let's plug those values in!

STEP 4

So, we have sin⁡(32.1∘)23.5=sin⁡(B)11.77\frac{\sin(32.1^{\circ})}{23.5} = \frac{\sin(B)}{11.77}23.5sin(32.1∘)​=11.77sin(B)​.

STEP 5

To get sin⁡(B)\sin(B)sin(B) by itself, we can multiply both sides of the equation by **11.77**: sin⁡(B)=11.77⋅sin⁡(32.1∘)23.5 \sin(B) = \frac{11.77 \cdot \sin(32.1^{\circ})}{23.5} sin(B)=23.511.77⋅sin(32.1∘)​

STEP 6

Calculating the right side gives us: sin⁡(B)≈0.2621 \sin(B) \approx 0.2621 sin(B)≈0.2621

STEP 7

Now, take the inverse sine (arcsin) of both sides to find BBB: B=arcsin⁡(0.2621) B = \arcsin(0.2621) B=arcsin(0.2621) B≈15.2∘ B \approx \textbf{15.2}^{\circ} B≈15.2∘

STEP 8

Since sin⁡(B)=sin⁡(180∘−B)\sin(B) = \sin(180^{\circ} - B)sin(B)=sin(180∘−B), there might be another possible angle BBB.Let's calculate the supplementary angle: 180∘−15.2∘=164.8∘180^{\circ} - 15.2^{\circ} = \textbf{164.8}^{\circ}180∘−15.2∘=164.8∘.

STEP 9

If B=164.8∘B = 164.8^{\circ}B=164.8∘, then A+B=32.1∘+164.8∘=196.9∘A + B = 32.1^{\circ} + 164.8^{\circ} = 196.9^{\circ}A+B=32.1∘+164.8∘=196.9∘.This is greater than 180∘180^{\circ}180∘, which is impossible for a triangle!So, there's only **one possible solution** for BBB.

STEP 10

The angles in a triangle add up to 180∘180^{\circ}180∘.We know A=32.1∘A = 32.1^{\circ}A=32.1∘ and B≈15.2∘B \approx 15.2^{\circ}B≈15.2∘, so C=180∘−A−BC = 180^{\circ} - A - BC=180∘−A−B.

STEP 11

C=180∘−32.1∘−15.2∘ C = 180^{\circ} - 32.1^{\circ} - 15.2^{\circ} C=180∘−32.1∘−15.2∘ C≈132.7∘ C \approx \textbf{132.7}^{\circ} C≈132.7∘

STEP 12

We can use the sine rule again: sin⁡(A)a=sin⁡(C)c\frac{\sin(A)}{a} = \frac{\sin(C)}{c}asin(A)​=csin(C)​.

STEP 13

We have sin⁡(32.1∘)23.5=sin⁡(132.7∘)c\frac{\sin(32.1^{\circ})}{23.5} = \frac{\sin(132.7^{\circ})}{c}23.5sin(32.1∘)​=csin(132.7∘)​.

STEP 14

Multiply both sides by ccc and 23.523.523.5, then divide by sin⁡(32.1∘)\sin(32.1^{\circ})sin(32.1∘): c=23.5⋅sin⁡(132.7∘)sin⁡(32.1∘) c = \frac{23.5 \cdot \sin(132.7^{\circ})}{\sin(32.1^{\circ})} c=sin(32.1∘)23.5⋅sin(132.7∘)​

STEP 15

c≈33.69 cm c \approx \textbf{33.69} \text{ cm} c≈33.69 cm

STEP 16

There is **one possible solution** to the triangle.The measurements for the remaining angles BBB and CCC and side ccc are as follows: B≈15.2∘B \approx 15.2^{\circ}B≈15.2∘, C≈132.7∘C \approx 132.7^{\circ}C≈132.7∘, and c≈33.69c \approx 33.69c≈33.69 cm.

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1. Use the sine rule to find angle $B$ .
2. Check for the ambiguous case.
3. Find angle $C$ .
4. Use the sine rule to find side $c$ .

The sine rule says that $\frac{\sin(A)}{a} = \frac{\sin(B)}{b}$ .
We know $a = \textbf{23.5 cm}$ , $b = \textbf{11.77 cm}$ , and $A = \textbf{32.1}^{\circ}$ .
Let's plug those values in!

So, we have $\frac{\sin(32.1^{\circ})}{23.5} = \frac{\sin(B)}{11.77}$ .

To get $\sin(B)$ by itself, we can multiply both sides of the equation by 11.77: $\sin(B) = \frac{11.77 \cdot \sin(32.1^{\circ})}{23.5}$

Calculating the right side gives us: $\sin(B) \approx 0.2621$

Now, take the inverse sine (arcsin) of both sides to find $B$ : $B = \arcsin(0.2621)$ $B \approx \textbf{15.2}^{\circ}$

Since $\sin(B) = \sin(180^{\circ} - B)$ , there might be another possible angle $B$ .
Let's calculate the supplementary angle: $180^{\circ} - 15.2^{\circ} = \textbf{164.8}^{\circ}$ .

If $B = 164.8^{\circ}$ , then $A + B = 32.1^{\circ} + 164.8^{\circ} = 196.9^{\circ}$ .
This is greater than $180^{\circ}$ , which is impossible for a triangle!
So, there's only one possible solution for $B$ .

The angles in a triangle add up to $180^{\circ}$ .
We know $A = 32.1^{\circ}$ and $B \approx 15.2^{\circ}$ , so $C = 180^{\circ} - A - B$ .

$C = 180^{\circ} - 32.1^{\circ} - 15.2^{\circ}$ $C \approx \textbf{132.7}^{\circ}$

We can use the sine rule again: $\frac{\sin(A)}{a} = \frac{\sin(C)}{c}$ .

We have $\frac{\sin(32.1^{\circ})}{23.5} = \frac{\sin(132.7^{\circ})}{c}$ .

Multiply both sides by $c$ and $23.5$ , then divide by $\sin(32.1^{\circ})$ : $c = \frac{23.5 \cdot \sin(132.7^{\circ})}{\sin(32.1^{\circ})}$

$c \approx \textbf{33.69} \text{ cm}$

There is one possible solution to the triangle.
The measurements for the remaining angles $B$ and $C$ and side $c$ are as follows: $B \approx 15.2^{\circ}$ , $C \approx 132.7^{\circ}$ , and $c \approx 33.69$ cm.