Answer:
- B = 80.3°, C = 59.7°, AB = 20.2
- B = 99.7°, C = 40.3°, AB = 15.1
Step-by-step explanation:
You want the solution to scalene triangle ABC with AC = 23, BC = 15, and ∠A = 40°.
Law of Sines
When two sides of a triangle are given, along with an angle not between them, the Law of Sines can be used to solve the triangle. It tells you ...
sin(B)/AC = sin(C)/AB = sin(A)/BC
When the given angle is opposite the shortest given side, there are generally 2 solutions.
Application
sin(B)/23 = sin(40°)/15 . . . . . fill in given values to find angle B
B = arcsin(23/15·sin(40°)) = 80.3° or 99.7°
Angle C will have the value that makes up the difference from 180°:
C = 180° -40° -{80.3°, 99.7°} = {59.7°, 40.3°}
The length of side AB can be found using the same Law of Sines proportion:
AB = BC·sin(C)/sin(A) = 15/sin(40°)·sin({59.7°, 40.3°})
AB = {20.2, 15.1}
The solutions are ...
- B = 80.3°, C = 59.7°, AB = 20.2
- B = 99.7°, C = 40.3°, AB = 15.1
Arguably, the second solution is not a "scalene" triangle. It is (nearly) isosceles. (See the comment below.)
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Additional comment
Since answers are requested to be rounded to tenths, both solutions are scalene triangles. If we rounded to integers, then only the first solution would be a scalene triangle.
