Question

Let ABC be a right-angled triangle having ZC = 90°.
Given that ZA is less than ZB. Also,
tan A+tan B+ tan^2 A+ tan^2B + tan^3A+ tan^3B= 70.
The degree measure of A lies in the interval
(A) (5,10) (B) (10,15) (C) (15, 20)
(D) (20,25)​

Answer 1

Given : ABC be a right-angled triangle ∠C = 90°  TanA + TanB  + Tan²A  + Tan²B  + Tan³A  + Tan³B = 70

To Find :  degree measure of A

Solution:

∠C = 90°

=> ∠A + ∠B = 90°

TanA + TanB  + Tan²A  + Tan²B  + Tan³A  + Tan³B = 70

=> TanA + CotA  + Tan²A  + Cot²A  + Tan³A  + Cot³A = 70

=> TanA + 1/TanA + Tan²A + 1/Tan²A  + Tan³A + 1/Tan²A = 70

=> (TanA + 1/TanA) + (TanA + 1/TanA)² - 2 + (TanA + 1/TanA) ³ - 3 (TanA + 1/TanA) = 70

=> -2(TanA + 1/TanA)  + (TanA + 1/TanA)²  + (TanA + 1/TanA) ³ = 72

(TanA + 1/TanA )(TanA + 1/TanA + 2)(TanA + 1/TanA - 1)  = 72

TanA + 1/TanA = 4

=> Tan²A - 4TanA + 1 = 0

=> Tan A =  2 ± √3

=> A = 75°  or  15°

A < B

Hence ∠A = 15°

measure of A = 15°  

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