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In triangle ABC, right-angled at B, if tan A = 1/√3 find the value of: (i) sin A cos C + cos A sin C (ii) cos A cos C – sin A sin C

 In triangle ABC, right-angled at B, if tan A = 1/√3 find the value of: (i) sin A cos C + cos A sin C (ii) cos A cos C – sin A sin C

Grade:12

1 Answers

Harshit Singh
askIITians Faculty 5963 Points
3 years ago
Dear Student

Let ΔABC in which∠B=90°

tan A =BC/AB = 1/√3
Let BC = 1k and AB =√3k,
Where k is the positive real number of the problemBy Pythagoras theorem in ΔABC we get: AC^2=AB^2+BC^2
AC^2=(√3k)^2+(k)^2
AC^2=3k^2+k^2
AC^2=4k^2
AC = 2k
Now find the values of cos A, Sin A
Sin A = BC/AC = 1/2
Cos A= AB/AC = √3/2
Then find the values of cos C and sin C
Sin C = AB/AC =√3/2
Cos C = BC/AC = 1/2
Now, substitute the values in the given problem

(i) sin A cos C + cos A sin C = (1/2) ×(1/2 )+ √3/2 ×√3/2 = 1/4 + 3/4 = 1

(ii) cos A cos C–sin A sin C = (√3/2 )(1/2)–(1/2) (√3/2 ) = 0


Thanks

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