Find the A : A = tan &#x2061;<!-- ⁡ --> 30 <

Celia Lucas

Celia Lucas

Answered question

2022-06-16

Find the A : A = tan 30 + tan 40 + tan 50 + tan 60 cos 20 = ?

Answer & Explanation

Quinn Everett

Quinn Everett

Beginner2022-06-17Added 23 answers

tan 40 + tan 50 = 1 cos 40 cos 50 = 1 cos 40 sin 40 = 2 sin ( 2 40 ) = 2 cos 10
tan 30 + tan 60 = 4 3
So we need
2 cos 10 + 4 3 cos 20 = 4 3 2 + cos 10 3 cos 20 cos 10
Now 3 2 + cos 10 = cos 30 + cos 10 = 2 cos 20 cos 10
arridsd9

arridsd9

Beginner2022-06-18Added 12 answers

Generalization:
tan ( 2 a 3 d ) + tan ( 2 a d ) + tan ( 2 a + d ) + tan ( 2 a + 3 d )
= sin ( 2 a 3 d + 2 a + 3 d ) cos ( 2 a 3 d ) cos ( 2 a + 3 d ) + sin ( 2 a d + 2 a + d ) cos ( 2 a d ) cos ( 2 a + d )
= sin 4 a { 2 cos ( 2 a 3 d ) cos ( 2 a + 3 d ) + 2 cos ( 2 a d ) cos ( 2 a + d ) } 2 cos ( 2 a 3 d ) cos ( 2 a d ) cos ( 2 a + d ) cos ( 2 a + 3 d )
= sin 4 a ( 2 cos 4 a + cos 6 d + cos 2 d ) 2 cos ( 2 a 3 d ) cos ( 2 a d ) cos ( 2 a + d ) cos ( 2 a + 3 d )
If 4 a = 90 , 2 a + 3 d = 90 ( 2 a 3 d ) ; 2 a + d = 90 ( 2 a d )
tan ( 45 3 d ) + tan ( 45 d ) + tan ( 45 + d ) + tan ( 45 + 3 d )
= cos 6 d + cos 2 d 2 cos ( 2 a 3 d ) cos ( 2 a d ) cos ( 2 a + d ) cos ( 2 a + 3 d )
= 4 cos 4 d cos 2 d 2 cos ( 2 a 3 d ) sin ( 2 a 3 d ) 2 sin ( 2 a d ) cos ( 2 a d )
= 4 cos 4 d cos 2 d sin ( 4 a 6 d ) sin ( 4 a 2 d )
= 4 cos 4 d cos 2 d cos 6 d cos 2 d  as  4 a = 90
= 4 cos 4 d cos 6 d  for  cos 2 d 0
Here d = 5

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