Complete right-triangle analysis from partial side and angle information.
Right triangles combine the Pythagorean theorem with sine, cosine, and tangent to recover missing parts. Given two sides, find the third; given a side and acute angle, trigonometry yields the rest.
Area equals half the product of legs: A = ab/2. Perimeter sums all three sides. Angle sums to 180° with one 90° angle, so acute angles complement to 90°.
Inverse trig functions recover angles from ratios. Calculator mode (degrees versus radians) must match your inputs to avoid scale errors.
Surveying, ramp design, and screen diagonal problems frequently use right triangle models. Sketch which angle is which before substituting into trig ratios.
Provide at least two pieces of information.
When two sides known without angles.
sin, cos, tan link sides and acute angles.
Multiply legs and divide by two.
Add all three side lengths.
Ensure sides satisfy triangle inequality.
c = √(a²+b²); A = ab/2; sinθ = opposite/hypotenuse
Pythagorean handles sides; trig handles angle-side links in right triangles.
A = (9×12)/2
Area = 54.
θ = arcsin(5/13)
θ ≈ 22.62°.
Other leg = √(625−49)
24.