a) find the unknowns
b) use the tangent to find the other leg; proceed as in (c)
c) use the arctangent to find the angle (if unknown); use the cosine and adjacent leg to find the hypotenuse. The other angle is the complement.
Step-by-step explanation:
a) "Solve a triangle" means "find the values of all the angles and side lengths." There are 6 measures in all. Three must be given, including at least one side (or hypotenuse) length, in order to solve the triangle.
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b) We interpret "side length" to mean the length of one leg, not the hypotenuse. If the hypotenuse is given the procedure is slightly different.
The given side will either be adjacent to or opposite the given angle. If it is adjacent, the opposite side is found from ...
Opposite = Tan( )×Adjacent
If the given side is opposite the angle, the adjacent side is found from ...
Adjacent = Opposite/Tan( )
Now, you have both opposite and adjacent sides, and one angle. To find the remaining measures, use the procedures in part (c).
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c) If neither angle is given, choose the longer side as the "adjacent" side and find the angle from ...
smaller acute angle = arctan(Opposite/Adjacent)
The other acute angle is the complement of the one you know.
Now, you have both acute angles, and both legs of the triangle.
Again, choosing the longer leg as "adjacent," use the cosine function to find the hypotenuse:
Hypotenuse = Adjacent/Cos( )
Of course, you can also find the hypotenuse using the Pythagorean theorem:
H = √(A² +O²)
Examples:
b) given a side 4 and an opposite angle of 57°.
Using the formula above:
adjacent = 4/tan(57°) = 2.5976
angle adjacent to 4 = 90° -57° = 33°
hypotenuse = 4/cos(33°) = 4.7695
Angles: 33°, 57°, 90°; sides: 2.5976, 4, 4.7695.
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c) given side lengths of 3 and 4
smaller acute angle = arctan(3/4) = 36.87°
larger acute angle = 90° -36.87° = 53.13°
hypotenuse = 4/cos(36.87°) = 5
Angles: 36.87°, 53.13°, 90°; sides: 3, 4, 5.
Note that in all cases, the full calculator precision should be maintained for all intermediate values. No numbers should be rounded until their final presentation as the solution to the problem.
Additional comment on this solution
For a given acute angle, you can also be given any of three sides: opposite, adjacent, or hypotenuse. To simplify our solution process here, we have assumed the hypotenuse is not a side given.
Basically, choose the relation in SOH CAH TOA that will have one unknown when you fill in the known values. Use that to find the unknown. Repeat until all are known, making use of the fact that acute angles are complementary, as needed. You can also use the Pythagorean theorem to find unknown sides when two are known.
