Find the value of cos 65 cos 55 cos 5
Answers
Tᴏ Fɪɴᴅ :-
- value of cos65° * cos55° * cos5° = ?
Fᴏʀᴍᴜʟᴀ ᴜsᴇᴅ :-
- cosA * cos(60 - A) * cos(60 + A) = (1/4) * cos3A
Sᴏʟᴜᴛɪᴏɴ :-
First Lets Try to Prove This direct Result :-
→ cosA * cos(60 - A) * cos(60 + A)
Multiply & Divide by 2 ,
→ (1/2)cosA [ 2 * cos(60 + A) * cos(60 - A) ]
using 2*cosA*cosB = cos(A+B) + cos(A - B)
→ (1/2)cosA [ cos(60 + A + 60 - A) + cos(60+A - 60 + A) ]
→ (1/2)cosA [ cos120° + cos2A ]
→ (1/2)cosA [ (-1/2) + cos2A ]
→ (-1/4)cosA + (1/2)cosA*cos2A
Again, Multiply & Divide by 2 ,
→ (-1/4)cosA + (1/4)[ 2 * cos2A * cosA ]
Again, using 2*cosA*cosB = cos(A+B) + cos(A - B)
→ (-1/4)cosA + (1/4)[ cos(2A + A) + cos(2A - A) ]
→ (-1/4)cosA + (1/4)[ cos3A + cosA ]
→ (-1/4)cosA + (1/4)cos3A + (1/4)cosA
→ (1/4)cos3A = RHS (Hence Proved).
____________________
Therefore,
→ cos65° * cos55° * cos5°
→ cos5° * cos55° * cos65°
→ cos5° * cos(60 - 5°) * cos(60 + 5°)
Comparing it with cosA * cos(60 - A) * cos(60 + A) we get,
→ (1/4) * cos(3*5)
→ (1/4) * cos15°
___________________
Now , Lets Find value of cos15° .
using cos(A - B) = cosA * cosB + sinA *sinB
→ cos15 = cos (45 - 30)
→ cos 15 = cos45 * cos30 + sin45 * sin30
→ cos 15 = (1/√2)* (√3/2) + (1/√2)* (1/2)
→ cos 15° = (√3 + 1 )/2√2
___________________
Hence,
→ Required Ans :- (1/4) * [ (√3 + 1 )/2√2 ]
→ Required Ans :- [ (√3 + 1)/8√2 ]
→ Required Ans :- [ √2(√3 + 1) / 16 ]