Advertisements
Advertisements
Question
Show that `cos pi/15 cos (2pi)/15 cos (3pi)/15 cos (4pi)/15 cos (5pi)/15 cos (6pi)/15 cos (7pi)/15 = 1/128`
Advertisements
Solution
`(pi/15 = 12^circ)`
L.H.S = cos 12° cos 24° cos 36° cos 48° cos 60° cos 72° cos 84° .....(1)
Consider (we know that)
cos A cos(60° + A) cos(60° – A)
= `cos "A" [cos^2 60^circ - sin^2"A"]`
= `cos "A"[1/4 - (1 - cos^2"A")]`
cos A cos(60° + A) cos(60° – A) = `1/4 cos 3"A"`
= `cos "A"[cos^2"A" - 3/4]`
= `(4cos^3 "A" - 3 cos "A")/4`
cos 12° cos 72° cos 48° = `1/4 cos 3(12^circ)`
= `1/4 cos 36^circ`
= `1/4[(sqrt(5) + 1)/4]`
Similarly cos 24° cos 84° cos 36° = `1/4 cos3 (12^circ)`
= `1/4 cos 72^circ`
= `1/4 cos(90^circ - 18^circ)`
= `1/4 sin 18^circ`
= `1/4[(sqrt(5) - 1)/4]`
(1) ⇒ L.H.S = `1/4[(sqrt(5) + 1)/4] * 1/4[(sqrt(5) - 1)/4] * 1/2`
= `1/4((sqrt(5) + 1)/4 * (sqrt(5) - 1)/4) * 1/2`
= `(5 - 1)/(128 xx 4)`
= `1/128`
APPEARS IN
RELATED QUESTIONS
Find the value of the trigonometric functions for the following:
cos θ = `- 1/2`, θ lies in the III quadrant
Find the value of the trigonometric functions for the following:
cos θ = `2/3`, θ lies in the I quadrant
If sin x = `15/17` and cos y = `12/13, 0 < x < pi/2, 0 < y < pi/2`, find the value of cos(x − y)
Find cos(x − y), given that cos x = `- 4/5` with `pi < x < (3pi)/2` and sin y = `- 24/25` with `pi < y < (3pi)/2`
Find the value of cos 105°.
Find the value of sin105°.
Prove that sin(π + θ) = − sin θ.
Expand cos(A + B + C). Hence prove that cos A cos B cos C = sin A sin B cos C + sin B sin C cos A + sin C sin A cos B, if A + B + C = `pi/2`
Prove that cos(A + B) cos C – cos(B + C) cos A = sin B sin(C – A)
Prove that `tan(pi/4 + theta) tan((3pi)/4 + theta)` = – 1
Find the value of cos 2A, A lies in the first quadrant, when tan A `16/63`
If θ is an acute angle, then find `sin (pi/4 - theta/2)`, when sin θ = `1/25`
Prove that cos 5θ = 16 cos5θ – 20 cos3θ + 5 cos θ
Express the following as a product
sin 75° sin 35°
Prove that `(sin(4"A" - 2"B") + sin(4"B" - 2"A"))/(cos(4"A" - 2"B") + cos(4"B" - 2"A"))` = tan(A + B)
If A + B + C = 180◦, prove that sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C
If A + B + C = 180°, prove that sin(B + C − A) + sin(C + A − B) + sin(A + B − C) = 4 sin A sin B sin C
Choose the correct alternative:
If cos 28° + sin 28° = k3, then cos 17° is equal to
Choose the correct alternative:
cos 1° + cos 2° + cos 3° + ... + cos 179° =
