Math, asked by saanchimahajan064, 9 months ago

Find the value of cos 65 cos 55 cos 5

Answers

Answered by RvChaudharY50
17

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°

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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

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Hence,

Required Ans :- (1/4) * [ (√3 + 1 )/2√2 ]

Required Ans :- [ (√3 + 1)/8√2 ]

Required Ans :- [ √2(√3 + 1) / 16 ]

Required Ans :- [ (√2 + √6) / 16 ]

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