Since a, b, c are in A.P 2 b = a + c We know the half-angle formula tan A 2 = = 2 b − b 2 b + b = 1 3 on simplification Therefore, tan (α + β) = tanα+tanβ 1−tanαtanβ t a n α + t a n β 1 − t a n α t a n β. Solved examples using the proof of tangent formula tan (α + β): 1. Find the values of tan 75°. Solution: tan 75° = tan ( 45° + 30°) = tan 45° + tan 30°/1 - tan 45° tan 30°. = 1 + 1/√3/1 - (1 . 1/√3) = √3 + 1/√3 - 1. www.mathportal.org 2 2 2 2 2 2 2 2 2 2 2 arctan 4 0 4 4 1 2 2 4 ln 4 0 4 2 4 2 4 0 2 ax b for ac b ac b ac b ax b b ac dx for ac b ax bx c b ac ax b b ac for ac b The tangent of the angle = the length of the opposite side the length of the adjacent side. So in shorthand notation: sin = o/h cos = a/h tan = o/a Often remembered by: soh cah toa. Example. Find the length of side x in the diagram below: The angle is 60 degrees. Now in triangle ADB, Tan 60 0 = AD/BD. = a√3/a = √3. Therefore, tan 60 degrees exact value is given by, Tan 60 0 =√3. In the same way, we can derive other values of tan degrees like 0 °, 30 °, 45 °, 90 °, 180 °, 270 ° and 360 °. Below is the trigonometry table, which defines all the values of tan along with other trigonometric ratios. iZyxS.

2 tan a tan b formula