A generalization of the Pythagorean theorem for any triangle. It relates one side to the other two sides and the included angle.
Law of Sines
sin(α)a=sin(β)b=sin(γ)c=2R
Direct proportionality between sides and the sines of their opposite angles. R is the circumradius.
Heron's Formula (Area)
A=s(s−a)(s−b)(s−c)
s=2a+b+c
Metrics & Radii
R=2sin(α)a(Circumradius)
r=sA(Inradius)
ha=a2A(Altitude on a)
ma=212b2+2c2−a2(Median on a)
1. Solving Triangles (Cases)
Solving a triangle means finding all its sides and angles from a minimum set of known values. There are several classic scenarios in analytical trigonometry: SSS (3 sides), SAS (2 sides and the included angle), and the ASA / AAS cases (two angles and one side). For these, we combine the Laws of Sines and Cosines. The sum of internal angles always imposes the universal constraint α+β+γ=180∘.
2. The Ambiguous Case (SSA)
When we know two sides (e.g., a,b) and a non-included angle (α opposite to a), we face the ambiguous case. The Law of Sines yields the equation sin(β)=absin(α).
If sin(β)>1: It is mathematically impossible. 0 solutions.
If sin(β)=1: The triangle is a right triangle. 1 solution.
If sin(β)<1: The arcsine returns an acute angle, but its supplement (obtuse) also has the same sine. There are 2 valid solutions geometrically (as long as the angle sum does not exceed 180°).
3. Engineering Applications (Right Triangle)
In construction, roads, and roofing, the right triangle is frequently parameterized by its Rise (vertical elevation or opposite leg) and Run (horizontal distance or adjacent leg). From these, the hypotenuse (length of the ramp/roof) and the slope percentage ((Rise/Run)×100) are derived. Complementary cut angles used in beam assemblies are also obtained.
4. Exact Special Angles Table
θ
0° (0 rad)
30° (π/6)
45° (π/4)
60° (π/3)
90° (π/2)
sin
0
1/2
√2/2
√3/2
1
cos
1
√3/2
√2/2
1/2
0
tan
0
√3/3
1
√3
∞
5. Physical Applications
Trigonometric functions are the cornerstone of classical and modern physics. For instance, Snell's Law uses the sine function to mathematically describe how light refracts when passing between different optical mediums (n1sinθ1=n2sinθ2). Similarly, in X-ray crystallography, Bragg's Diffraction determines atomic-level structures and distances using the sine of the angle of incidence (nλ=2dsinθ). Furthermore, in Projectile Motion, sine and cosine are constantly used to decompose velocity vectors and calculate trajectories.