Standard notation

Sides \(a, b, c\) opposite angles \(A, B, C\). Angles sum to 180°:

$$ A + B + C = 180^\circ $$

Law of Sines

Use when you know two angles + any side (AAS, ASA) or two sides + a non-included angle (SSA, careful — ambiguous case below).

$$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R $$

(\(R\) is the circumradius — the common ratio equals the diameter of the circumscribed circle.)

Example (AAS). \(A = 40°\), \(B = 70°\), \(a = 8\). Then \(C = 70°\), and:

$$ b = a \cdot \frac{\sin B}{\sin A} = 8 \cdot \frac{\sin 70^\circ}{\sin 40^\circ} \approx 11.7 $$ 
import math
a = 8
b = a * math.sin(math.radians(70)) / math.sin(math.radians(40))
print(round(b, 2))  # 11.7

Law of Cosines

Use for SSS (three sides → angles) or SAS (two sides + included angle → third side). It generalizes Pythagoras:

$$ c^2 = a^2 + b^2 - 2ab\cos C $$

Cyclic versions hold for the other sides. When \(C = 90°\), \(\cos C = 0\) and you recover \(c^2 = a^2 + b^2\).

Example (SAS). \(a = 5\), \(b = 7\), \(C = 60°\):

$$ c^2 = 25 + 49 - 2(5)(7)\cos 60^\circ = 74 - 35 = 39 \quad\Rightarrow\quad c \approx 6.24 $$

Example (SSS → angle). Sides 3, 4, 5 — is it right? Solve for the angle opposite side 5:

$$ \cos C = \frac{3^2 + 4^2 - 5^2}{2\cdot 3 \cdot 4} = 0 \quad\Rightarrow\quad C = 90^\circ $$

Which law when?

Known Use
AAS, ASA Law of Sines
SAS, SSS Law of Cosines (then Sines for remaining)
SSA Law of Sines, but check the ambiguous case (0, 1, or 2 triangles)
AAA Nothing — similar triangles; angles alone never fix size

Solver sketch:

import math

def law_of_cos_side(a, b, C_deg):
    C = math.radians(C_deg)
    return math.sqrt(a*a + b*b - 2*a*b*math.cos(C))

def law_of_sines_side(a, A_deg, B_deg):
    return a * math.sin(math.radians(B_deg)) / math.sin(math.radians(A_deg))

def angle_from_sss(a, b, c):
    # angle opposite side c
    cosC = (a*a + b*b - c*c) / (2*a*b)
    cosC = max(-1.0, min(1.0, cosC))  # clamp for float safety
    return math.degrees(math.acos(cosC))

print(round(law_of_cos_side(5, 7, 60), 2))  # 6.24
print(round(angle_from_sss(3, 4, 5), 2))    # 90.0
function lawOfCosSide(a, b, cDeg) {
  const c = cDeg * Math.PI / 180;
  return Math.sqrt(a*a + b*b - 2*a*b*Math.cos(c));
}
console.log(lawOfCosSide(5, 7, 60).toFixed(2)); // 6.24

Watch out

  • Degree/radian mix-ups — convert before sin/cos, convert back after asin/acos.
  • SSA ambiguity — e.g. side-angle-side with the angle not included can yield two valid triangles. If your solver finds \(\sin B > 1\), no triangle exists; if \(\sin B\) gives \(B\) acute, \(180° − B\) may also work.
  • Float clamp — always clamp cos inputs to \([−1, 1]\) before acos, or rounding error throws NaN / ValueError.

Practice

  1. Triangle: \(A = 50°\), \(B = 60°\), \(c = 10\) (side between them). Find \(C\) then sides \(a, b\).
  2. Sides 7, 8, 9. Find the largest angle. (Answer ≈ 73.4°.)
  3. Two GPS points form SAS: legs 3 km and 4 km with 90° between. How far apart are the endpoints? (Sanity: 3-4-5.)
  4. Code it: extend the solver above to take SSS and return all three angles; assert they sum to 180°.

Next: Graphs of Trig Functions — turning these values into waves.