Answer:
- 2 possible solutions
- D ≈ 49°, E ≈ 81°, d ≈ 61
- D ≈ 31°, E ≈ 99°, d ≈ 42
Explanation:
Given ∆DEF with e = 80, f = 62, and F = 50°, you want the possible solutions to the triangle.
Law of sines
The law of sines tells you ...
sin(E)/e = sin(F)/f
E = arcsin(e/f·sin(F))
The value of e/f·sin(F) is approximately 0.98844444273, which gives two possibilities for E:
E = arcsin(0.98844444273) ≈ 81.3°
E = 180° -arcsin(0.98844444273) ≈ 98.7°
Third angle
The third angle (D) of the triangle will be the value that makes the sum of angles be 180°.
D = 180° -F -E = 180° -50° -{81.3°, 98.7°’
D = {48.7°, 31.3°}
Third side
The law of sines can be used again to find the length of the third side:
d/sin(D) = f/sin(F)
d = f·sin(D)/sin(F)
d = 62·sin({48.7°, 31.3°})/sin(50°) ≈ {60.8, 42.0}
The 2 possible solutions to the triangle are ...
- D = 48.7°, E = 81.3°, d = 60.8
- D = 31.3°, E = 98.7°, d = 42.0
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Additional comment
When the given angle is opposite the shorter of two given sides, as here, there will usually be 2 solutions to the triangle. Exceptions are (a) when the triangle is a right triangle, or (b) the triangle cannot exist.
The two solutions are shown in the diagrams of the last two attachments.
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