cos 6°
sin24° cos72° =
Answers
Answer:
Step-by-step explanatiocdn:
Answer:
1/8
Step-by-step explanation:
Q. Evaluate: Cos6° × sin24° × cos72°
Soln:
Cos6° × sin24° × cos72°
= (1/2) * 2 Cos6° × sin24° × cos72°
= (1/2) [sin(6+24) – sin(6-24)] * cos72
= (1/2) [sin30 – sin(-18)] * cos72
= (1/2) [1/2 + sin18] * cos72 …….eqn(i)
We know,
Sin72 = 2 sin36 * cos36
cos(90-72) = 2 (2 sin18 * cos18) * (1 – 2sin218)
cos18 = 4sin18.cos18 * (1 – 2sin218)
1 = 4sin18 * (1 – 2sin218) …………eqn(ii)
Let sin18 = x, then
1 = 4x * (1 – 2x2)
8x3 – 4x + 1 = 0 [Hint: use synthetic division formula to factorize.]
(2x – 1)(4x2 + 2x – 1) = 0
Here,
x = 1/2
sin18 = 1/2 is not possible .
4x2 + 2x – 1 = 0 gives
x = (Ö5 – 1)/4 (Hint: compare given equation with ax2 + bx + c = 0 )
sin18 = (Ö5 – 1)/4
put value of sin18 in eqn(i)
= (1/2) [1/2 + (Ö5 – 1)/4] * cos72
= (1/2) [1/2 + (Ö5 – 1)/4] * sin(90 – 72)
= (1/2) [1/2 + (Ö5 – 1)/4] * sin18
= (1/2) [1/2 + (Ö5 – 1)/4] * (Ö5 – 1)/4
= (1/2) [(Ö5 + 1)/4] * (Ö5 – 1)/4
= (5 – 1)/32
= 4/32
= 1/ 8 Ans.