Math / Geometry

QuestionFind the unknown angles in triangle $A B C$ for the following triangle if it exists. $\mathrm{C}=48^{\circ} 20^{\prime}, \mathrm{b}=26.5 \mathrm{~m}, \mathrm{c}=31.1 \mathrm{~m}$
Select the correct choice below, and, if necessary, fill in the answer boxes to complete your choice. A. There is only one possible set of remaining angles. The measurements for the remaining angles are $A=$ $\square$ ${ }^{\circ}$ ' and $B=$ $\square$ ${ }^{\circ}$ ${ }^{\prime}$ . '. (Do not round until the final answers. Then round to the nearest whole number as needed.) B. There are two possible sets of remaining angles. The measurements for when $B$ is larger are $A_{1}=$ $\square$ ${ }^{\circ}$ $\square$ ' and $\mathrm{B}_{1}=$ $\square$ ${ }^{\circ}$ '. - The measurements for when $B$ is smaller are $A_{2}=$ $\square$ ${ }^{\circ}$ (Do not round until the final answers. Then round to the nearest whole number as needed.) C. No such triangle exists.

Studdy Solution

STEP 1

1. We are given a triangle $\triangle ABC$ with angle $C = 48^\circ 20'$ .
2. Side $b = 26.5 \, \text{m}$ and side $c = 31.1 \, \text{m}$ .
3. We need to determine the remaining angles $A$ and $B$ .
4. We will use the Law of Sines to find the unknown angles.

STEP 2

1. Check the possibility of the triangle's existence using the given sides and angle.
2. Use the Law of Sines to find angle $B$ .
3. Determine angle $A$ using the angle sum property of triangles.
4. Check for the possibility of a second triangle (ambiguous case).
5. Conclude with the correct choice among the given options.

STEP 3

Check if the triangle exists using the given sides and angle. According to the triangle inequality theorem, the sum of any two sides must be greater than the third side. Here, we are given two sides and an angle, so we will use the Law of Sines to check if a valid angle $B$ can be found.

STEP 4

Apply the Law of Sines:
$\frac{\sin B}{b} = \frac{\sin C}{c}$
Substitute the known values:
$\frac{\sin B}{26.5} = \frac{\sin(48^\circ 20')}{31.1}$
Calculate $\sin(48^\circ 20')$ and solve for $\sin B$ :
$\sin B = \frac{26.5 \cdot \sin(48^\circ 20')}{31.1}$

STEP 5

Calculate $\sin(48^\circ 20')$ using a calculator:
$\sin(48^\circ 20') \approx 0.748$
Substitute back to find $\sin B$ :
$\sin B = \frac{26.5 \cdot 0.748}{31.1} \approx 0.637$
Find $B$ using the inverse sine function:
$B = \sin^{-1}(0.637) \approx 39^\circ 30'$

STEP 6

Use the angle sum property of triangles to find angle $A$ :
$A = 180^\circ - C - B$
Substitute the known values:
$A = 180^\circ - 48^\circ 20' - 39^\circ 30'$
Calculate $A$ :
$A = 92^\circ 10'$

STEP 7

Check for the possibility of a second triangle (ambiguous case). Since $\sin B = 0.637$ , there could be another angle $B'$ such that:
$B' = 180^\circ - B = 180^\circ - 39^\circ 30' = 140^\circ 30'$
Check if this leads to a valid triangle:
$A' = 180^\circ - C - B' = 180^\circ - 48^\circ 20' - 140^\circ 30'$
Calculate $A'$ :
$A' = -8^\circ 50'$
Since $A'$ is not a valid angle, there is no second triangle.

STEP 8

Conclude with the correct choice among the given options. Since there is only one valid set of angles, the correct choice is:
A. There is only one possible set of remaining angles. The measurements for the remaining angles are $A = 92^\circ 10'$ and $B = 39^\circ 30'$ .
The solution is:
A. $A = 92^\circ 10'$ , $B = 39^\circ 30'$ .

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Studdy Solution

STEP 1

1. We are given a triangle $\triangle ABC$ with angle $C = 48^\circ 20'$ .
2. Side $b = 26.5 \, \text{m}$ and side $c = 31.1 \, \text{m}$ .
3. We need to determine the remaining angles $A$ and $B$ .
4. We will use the Law of Sines to find the unknown angles.

STEP 2

STEP 3

Check if the triangle exists using the given sides and angle. According to the triangle inequality theorem, the sum of any two sides must be greater than the third side. Here, we are given two sides and an angle, so we will use the Law of Sines to check if a valid angle $B$ can be found.

STEP 4

Apply the Law of Sines:
$\frac{\sin B}{b} = \frac{\sin C}{c}$
Substitute the known values:
$\frac{\sin B}{26.5} = \frac{\sin(48^\circ 20')}{31.1}$
Calculate $\sin(48^\circ 20')$ and solve for $\sin B$ :
$\sin B = \frac{26.5 \cdot \sin(48^\circ 20')}{31.1}$

STEP 5

Calculate $\sin(48^\circ 20')$ using a calculator:
$\sin(48^\circ 20') \approx 0.748$
Substitute back to find $\sin B$ :
$\sin B = \frac{26.5 \cdot 0.748}{31.1} \approx 0.637$
Find $B$ using the inverse sine function:
$B = \sin^{-1}(0.637) \approx 39^\circ 30'$

STEP 6

Use the angle sum property of triangles to find angle $A$ :
$A = 180^\circ - C - B$
Substitute the known values:
$A = 180^\circ - 48^\circ 20' - 39^\circ 30'$
Calculate $A$ :
$A = 92^\circ 10'$

STEP 7

STEP 8

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Studdy Solution

STEP 1

STEP 2

STEP 3

Check if the triangle exists using the given sides and angle. According to the triangle inequality theorem, the sum of any two sides must be greater than the third side. Here, we are given two sides and an angle, so we will use the Law of Sines to check if a valid angle B B B can be found.

STEP 4

STEP 5

STEP 6

STEP 7

STEP 8

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Check if the triangle exists using the given sides and angle. According to the triangle inequality theorem, the sum of any two sides must be greater than the third side. Here, we are given two sides and an angle, so we will use the Law of Sines to check if a valid angle $B$ can be found.