Triangle identities · trigonometric identity
a = b·cos C + c·cos B — Projection rule
When to use it
A side has to be split into the two projections of the other sides — the step that turns a pair of sine-rule ratios into a single expression for a side, and the shortcut in proofs of the tangent rule. It is a relation between sides and angles, not a solver for the ambiguous case.
Why it is true
Drop the perpendicular from C onto AB. It cuts side c into the pieces b·cos A and a·cos B; writing the same split for the other two sides and adding the two pieces that make up a gives a = b·cos C + c·cos B.
The full line-by-line version is on the proof page for projection rule; the “how would I find this myself” version is in the derivation.
Where it comes from
combine the sine rule with the angle sum. A second route that does not draw an altitude: start from the ratio the sine rule gives and reduce the angle sum with the sine addition formula.
- b·cos C + c·cos B = R(sin B cos C + sin C cos B)Replace b and c by the sine rule form, with R the common ratio a/sin A. · Law of sines
- = R·sin(B + C)Sine addition formula. · Sine of a sum
- = R·sin A = aB + C = π − A from the angle sum, and sin(π − A) = sin A. · Angles of a triangle add to 180°
Worked examples
No worked example is written for this identity yet —the examples index lists what is covered, and the pattern below shows the identity working at real angles.
Checked on these triangles
Two sides of the identity evaluated with the same engine that runs the verifier. A single line disagreeing would break the build.
| Triangle (A) | Left side | Right side | Agree |
|---|---|---|---|
| 7.5° | 1.621074 | 1.621074 | yes |
| 15° | 0.778211 | 0.778211 | yes |
| 18° | 1.1532 | 1.1532 | yes |
| 22.5° | 1.997561 | 1.997561 | yes |
| 30° | 3.204091 | 3.204091 | yes |
| 37° | 2.504847 | 2.504847 | yes |
| 45° | 2.277435 | 2.277435 | yes |
| 53° | 3.046822 | 3.046822 | yes |
| 60° | 3.966343 | 3.966343 | yes |
| 67.5° | 3.620247 | 3.620247 | yes |
| 75° | 3.622348 | 3.622348 | yes |
| 90° | 4.816443 | 4.816443 | yes |
| 120° | 5.537542 | 5.537542 | yes |
| 135° | 5.497117 | 5.497117 | yes |
| 150° | 6.010828 | 6.010828 | yes |
The letters are not free
a, b, c and A, B, C here are one triangle's sides and angles: A + B + C = 180° and the sides follow the sine rule. Feed the same six letters with random values and the equation fails — that is a property of the formula, not a bug in the tool.
The mistake students make
Reading it as a = b·cos B + c·cos C, i.e. pairing each side with its own adjacent angle. Each cosine must be taken at the angle at the OTHER end of side a, otherwise the two pieces are not the two projections of a.
Try it
- Verify this identity — both sides are pre-filled; change one character and watch the counterexample appear.
- Ask “which identity should I use?” about the left-hand side — this identity is what the chooser should rank first.
- Cheat sheet entry for triangle identities.
Related identities
- a/sin A = b/sin BLaw of sinesdetailsproof
- a² = b² + c² − 2bc·cos ALaw of cosinesdetailsproof
- sin(A + B) = sin A cos B + cos A sin BSine of a sumdetailsproof
Category hub: Triangle identities · all identities: /identities