Lesson 10 of 13 · 8 min

Tangent and cotangent of sums

NCERT §3.4

Meera's spoke makes angle A with the horizontal and a friend's spoke is a further B round. If we know how steep each angle is on its own, tan A and tan B, how steep is the angle A + B?

The lesson in notes

In short

tan (A + B) = (tan A + tan B)/(1 − tan A tan B), valid when none of A, B and A + B is an odd multiple of π/2, so that cos A, cos B and cos (A + B) are all non-zero.

It comes from dividing sin (A + B) = sin A cos B + cos A sin B by cos (A + B) = cos A cos B − sin A sin B, then dividing the top and bottom by cos A cos B.

tan (A − B) = (tan A − tan B)/(1 + tan A tan B), obtained by putting −B for B.

cot (A + B) = (cot A cot B − 1)/(cot B + cot A), valid when none of A, B and A + B is a multiple of π; and cot (A − B) = (cot A cot B + 1)/(cot B − cot A), valid when none of A, B and A − B is a multiple of π.

Worked value: tan 105° = tan (60° + 45°) = (√3 + 1)/(1 − √3 × 1) = (√3 + 1)/(1 − √3); multiplying top and bottom by (1 + √3) gives (4 + 2√3)/(−2) = −(2 + √3).

With A = π/4, where tan A = 1: tan (π/4 + B) = (1 + tan B)/(1 − tan B) and tan (π/4 − B) = (1 − tan B)/(1 + tan B).

The formula can also identify an angle: if tan A = 1/2 and tan B = 1/3 with A and B acute, then tan (A + B) = (5/6)/(1 − 1/6) = 1, and since 0 < A + B < π, A + B = π/4.