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Projection Formula: Derivation and Examples Explained

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The projection formula says that each side of a triangle equals the sum of the projections of the other two sides onto it: a = b cos C + c cos B. A projection is like a shadow — the length of a segment measured along the direction of another line.

This matters because it shows how a whole side is built from the other two sides — the same idea behind splitting forces and velocities into parts in physics.

Key Concepts

  • Projection of a side: the projection of side b onto side a is b cos C (the angle between a and b is C), and the projection of side c onto side a is c cos B.
  • Derivation by altitude: drop a perpendicular from vertex A to side a (BC), meeting it at D. In right triangle ABD, the leg BD = c cos B; in right triangle ACD, the leg DC = b cos C. Since BD + DC = a, we get a = c cos B + b cos C.
  • Cyclic versions: the same reasoning from each vertex gives b = c cos A + a cos C and c = a cos B + b cos A.
  • Obtuse triangles: when an angle is obtuse, one projection falls outside the side and counts as negative. The formula still holds exactly.

Formulas

  • Projection formula: a = b cos C + c cos B; b = c cos A + a cos C; c = a cos B + b cos A.
  • Assumptions: A, B, C are the interior angles of a triangle (summing to 180°) and sides a, b, c are opposite them. Length units are consistent on both sides; angles are in degrees. Projections are signed when an angle is obtuse.

Worked Example

In a triangle, b = 10 cm, c = 8 cm, B = 60° and C = 45°. Find side a.

  1. Write the formula: a = b cos C + c cos B.
  2. Substitute: a = 10 × cos 45° + 8 × cos 60°.
  3. cos 45° ≈ 0.7071 and cos 60° = 0.5, so a ≈ 10 × 0.7071 + 8 × 0.5 = 7.071 + 4 = 11.071.
  4. a ≈ 11.07 cm.

Common Misconceptions

  • Mixing it up with the cosine rule: the cosine rule multiplies lengths (c² = a² + b² − 2ab cos C), while the projection formula adds projections (a = b cos C + c cos B). They are related but different statements.
  • Acute-only thinking: the formula does not fail for obtuse triangles — the projection falling outside the side is simply negative, and the sum still works.
  • Wrong angle pairing: the projection of side b uses cos C, the angle between b and the side you project onto — not b's own opposite angle.

Real-World Applications

  • Physics: resolving a force into components along two directions — each component is a projection, exactly as the formula describes.
  • Navigation: finding how much of a ship's velocity carries it east versus north — the east and north speeds are projections of the total velocity.
  • Mathematical proofs: the projection formula is a stepping stone to proving the sine rule and other triangle identities.

The projection formula a = b cos C + c cos B shows that side a is exactly the sum of the shadows the other two sides cast along it. Drop an altitude to see why, pair each side with the angle beside it, and remember obtuse triangles work too, with signed projections.

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