Answer:
Two solutions:
- a = 15/2, c = (5√13)/2
- a = (15√13)/13, c = (10√13)/13
Explanation:
Given right ΔABC, with tan(A) = 3/2 and b = 5, you want the exact lengths of sides a and c.
Solving a triangle
To completely solve a triangle, you need to know at least one side, at least one angle, and one other side or angle. Here, we're given one side and one angle, so cannot solve the triangle using the given information.
We know another angle is 90°, but we don't know which one.
Often, side c is the one designated as the hypotenuse, but that is not necessarily the case. So, there are two solutions.
C is the right angle
This case corresponds to the red triangle shown in the attachment.
In this triangle, the leg adjacent to angle A is side b, so we have ...
Tan = Opposite/Adjacent . . . . . trig relation
3/2 = a/5
a = 15/2
The Pythagorean theorem can be used to find the length of side c.
c² = a² +b²
c² = (15/2)² +5² = 225/4 +25 = 325/4
c = √(325/4)
c = (5/2)√13
B is the right angle
This case corresponds to the blue triangle shown in the attachment.
The given tangent of angle A tells us the ratio ...
Tan = Opposite/Adjacent
3/2 = a/c
a = 3/2c . . . . . multiply by c
The Pythagorean theorem tells us ...
b² = a² +c²
5² = (3/2c)² +c² = 13/4c² . . . . . . substitute for b and a
c² = (4/13)(5²) = (10/13)²·13 . . . . . . multiply by 13/13
c = (10√13)/13 . . . . . . . . . . . . square root
a = 3/2c = (15√13)/13 . . . . . . find 'a'