9514 1404 393
Answer:
- a = 24; α = 73.7°; β = 16.3°
- c = 65; α = 14.3°; β = 75.7°
- a = b = 102.53; β = 45°
- b = 22.52; c = 26; β = 60°
Step-by-step explanation:
The relevant relations are provided by the Pythagorean theorem and the definitions of the trig ratios. For finding angles, inverse trig functions are required, generally meaning a calculator is involved.
Pythagorean theorem: a² +b² = c² for legs a, b, and hypotenuse c.
Trig definitions: "SOH CAH TOA"
- Sin = Opposite/Hypotenuse
- Cos = Adjacent/Hypotenuse . . . . α = arccos(b/c)
- Tan = Opposite/Adjacent . . . . . . . α = arctan(a/b)
It is also helpful to know the side ratios of the "special" right triangles:
isosceles right triangle: 1 : 1 : √2
30°-60°-90° right triangle: 1 : √3 : 2
__
<u>Triangle 1</u>:
a = √(25² -7²) = √576 = 24
α = arccos(7/25) ≈ 73.7°
β = 90° -73.7° = 16.3°
<u>Triangle 2</u>:
c = √(16² +63²) = √4225 = 65
α = arctan(16/63) ≈ 14.3°
β = 90° -14.3° = 75.7°
<u>Triangle 3</u>:
The hypotenuse is given = √2 times the other sides.
a = b = c/√2 = 72.5√2 ≈ 102.53 . . . . isosceles right triangle
β = 90° -45° = 45°
<u>Triangle 4</u>:
The short side is given, corresponding to 1 ratio unit.
b = 13×√3 ≈ 22.52 . . . . 30°-60°-90° special triangle
c = 13×2 = 26
β = 90° -30° = 60°
_____
<em>Additional comments</em>
On-line triangle solver calculators are available, easily found with a search.
The above-mentioned right triangles are "special" in that for an angle with an integer number of degrees, at least one of the trig functions is rational. These are the only triangles with that characteristic.
