Advertisements
Advertisements
Question
Show that sin 12° sin 48° sin 54° = `1/8`
Solution
sin 12° . sin 48° . sin 54° = sin 48° . sin 12° . sin(90° – 36°)
= `1/2` [cos (48° – 12°) – cos (48° + 12°)] cos 36°
= `1/2` [cos 36° – cos 6o°] cos 36°
= `1/2 [cos 36^circ - 1/2] cos 36^circ`
= `1/2 [cos^2 36^circ - 1/2 cos 36^circ]`
= `1/2 [((sqrt(5) + 1)/4)^2 - 1/2((sqrt(5) + 1)/4)]`
= `1/2[(5 + 2sqrt(5) + 1)/16 - ((sqrt(5) + 1)/8)]`
= `1/16 [(5 + 2sqrt(5) + 1)/2 - (sqrt(5) + 1)]`
= `1/16 [(6 + 2sqrt(5) - 2sqrt(5) - 2)/2]`
= `1/16 xx 4/2`
= `1/8`
APPEARS IN
RELATED QUESTIONS
Find the values of cot(660°)
Find the value of the trigonometric functions for the following:
cos θ = `- 1/2`, θ lies in the III quadrant
If sin A = `3/5` and cos B = `9/41, 0 < "A" < pi/2, 0 < "B" < pi/2`, find the value of cos(A – B)
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°.
Prove that cos(30° + x) = `(sqrt(3) cos x - sin x)/2`
Prove that sin(30° + θ) + cos(60° + θ) = cos θ
Prove that cos(A + B) cos(A – B) = cos2A – sin2B = cos2B – sin2A
Find the value of tan(α + β), given that cot α = `1/2`, α ∈ `(pi, (3pi)/2)` and sec β = `- 5/3` β ∈ `(pi/2, pi)`
If θ + Φ = α and tan θ = k tan Φ, then prove that sin(θ – Φ) = `("k" - 1)/("k" + 1)` sin α
Find the value of cos 2A, A lies in the first quadrant, when cos A = `15/17`
Prove that `32(sqrt(3)) sin pi/48 cos pi/48 cos pi/24 cos pi/12 cos pi/6` = 3
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`
Prove that `(sin 4x + sin 2x)/(cos 4x + cos 2x)` = tan 3x
If A + B + C = 180°, prove that sin2A + sin2B − sin2C = 2 sin A sin B cos C
If A + B + C = 180°, prove that `tan "A"/2 tan "B"/2 + tan "B"/2 tan "C"/2 + tan "C"/2 tan "A"/2` = 1
If A + B + C = 180°, prove that sin A + sin B + sin C = `4 cos "A"/2 cos "B"/2 cos "C"/2`
If x + y + z = xyz, then prove that `(2x)/(1 - x^2) + (2y)/(1 - y^2) + (2z)/(1 - z^2) = (2x)/(1 - x^2) (2y)/(1 - y^2) (2z)/(1 - z^2)`
If ∆ABC is a right triangle and if ∠A = `pi/2` then prove that cos B – cos C = `- 1 + 2sqrt(2) cos "B"/2 sin "C"/2`
