QuestionTwo sides and an angle are given below. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results.
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
(Type an integer or decimal rounded to two decimal places as needed.)
Studdy Solution
STEP 1
1. We are given two sides and a non-included angle (SSA condition).
2. We need to determine if the given information results in one triangle, two triangles, or no triangle.
3. If a triangle or triangles exist, we need to solve for all unknown sides and angles.
STEP 2
1. Use the Law of Sines to determine the possible number of triangles.
2. Determine the number of triangles (one, two, or none).
3. Solve for the unknown sides and angles of any resulting triangle(s).
STEP 3
Use the Law of Sines to find angle . The Law of Sines states:
Substitute the known values:
Solve for :
Calculate :
STEP 4
Determine the possible values for angle . Since , we find using the inverse sine function:
Check for a possible second triangle. Since the sine function is positive in the first and second quadrants, another possible angle is:
Check if both angles and can form a triangle with angle :
1. For :
$ A + B = 70^\circ + 38.79^\circ = 108.79^\circ < 180^\circ
\]
This is valid.
2. For :
$ A + B' = 70^\circ + 141.21^\circ = 211.21^\circ > 180^\circ
\]
This is not valid.
Thus, only one triangle is possible.
STEP 5
Solve the triangle with .
Find angle :
Use the Law of Sines to find side :
Substitute the known values:
Solve for :
Calculate :
The given information results in one triangle with the following solution:
- Angle
- Angle
- Side
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