Advertisements
Advertisements
प्रश्न
Prove the following:
sin 36° = `(sqrt(10 - 2sqrt(5)))/4`
Advertisements
उत्तर
Let θ = 18°
∴ 5θ = 90°
∴ 2θ + 3θ = 90°
∴ 2θ = 90° – 3θ
∴ sin 2θ = sin (90° – 3θ)
∴ sin 2θ = cos 3θ
∴ 2 sin θ cos θ = 4 cos3θ – 3 cos θ
∴ 2 sin θ = 4 cos2θ – 3 ...[∵ cos θ ≠ 0]
∴ 2 sin θ = 4 (1 – sin2θ) – 3
∴ 2 sin θ = 1 – 4 sin2θ
∴ 4 sin2θ + 2 sin θ – 1 = 0
∴ sin θ = `(-2 ± sqrt(4 + 16))/8`
= `(-2 ± 2sqrt(5))/8`
= `(-1 ± sqrt(5))/4`
Since, sin 18° > 0
∴ sin 18° = `(sqrt(5) - 1)/4`
∴ cos218° = 1 – sin218°
= `1 - ((sqrt(5) - 1)/4)^2`
= `1 - ((5 - 2sqrt(5) + 1)/16)`
= `(16 - 5 + 2sqrt(5) - 1)/16`
= `(10 + 2sqrt(5))/16`
∴ cos 18° = `sqrt(10 + 2sqrt(5))/4` ...[∵ 18° is an acute angle]
∴ sin 36° = 2 sin 18° · cos 18°
= `2 xx (sqrt(5) - 1)/4 xx sqrt(10 + 2sqrt(5))/4`
= `(sqrt((sqrt(5) - 1)^2) xx sqrt(10 + 2sqrt(5)))/8`
= `sqrt((6 - 2sqrt(5))(10 + 2sqrt(5)))/8`
= `(sqrt(60 + 12sqrt(5) - 20sqrt(5) - 20))/8`
= `sqrt(40 - 8sqrt(5))/8`
= `(2sqrt(10 - 2sqrt(5)))/8`
∴ sin 36° = `(sqrt(10 - 2sqrt(5)))/4`
APPEARS IN
संबंधित प्रश्न
Find the value of:
sin 15°
Find the values of:
tan 105°
Find the value of :
sin (495°)
Find the value of :
cos (600°)
Find the value of :
tan 225°
Find the value of :
tan (– 690°)
Find the value of :
cot (– 1110°)
Prove the following:
`cos((3pi)/2 + x) cos(2pi + x)[cot((3pi)/2 - x) + cot(2pi + x)]` = 1
Prove the following:
`("cosec"(90^circ - x)sin(180^circ - x)cot(360^circ - x))/(sec(180^circ + x)tan(90^circ + x)sin(-x))` = 1
Prove the following:
cosθ + sin (270° + θ) − sin (270° − θ) + cos (180° + θ) = 0
Select the correct option from the given alternatives :
Let 0 < A, B < `pi/2` satisfying the equation 3 sin2A + 2 sin2B = 1 and 3 sin 2A − 2 sin 2B = 0 then A + 2B is equal to ______
Prove the following:
tan 20° tan 80° cot 50° = `sqrt(3)`
Prove the following:
cos 36° = `(sqrt(5) + 1)/4`
Prove the following:
`tan pi/8 = sqrt(2) - 1`
Prove the following:
sin47° + sin61° − sin11° − sin25° = cos7°
If θ = `(17π)/3` then, tan θ – cot θ = ______.
`(1 - 2[cos 60^circ - cos 80^circ])/(2 sin 10^circ)` = ______.
The value of sin(– 1125°) is ______.
The value of `2 cot^2(π/6) + 4 tan^2(π/6) - 3 "cosec"(π/6)` is ______.
The value of `(cot 54^circ)/(tan 36^circ) + (tan 20^circ)/(cot 70^circ)` is ______.
The value of cos 480° sin 150° + sin 600° cos 390° is ______.
In a ΔABC, if ∠A = `π/2`, then cos2 B + cos2 C is equal to ______.
If ΔABC is a right angled at C, then tan A + tan B is equal to ______.
If `sin A - sqrt(6) cos A = sqrt(7) cos A`, then `cos A + sqrt(6) sin A` is equal to ______.
If sin A + sin B + sin C = 3, then cos A + cos B + cos C is equal to ______.
In a ΔPQR, if 3 sin P + 4 cos Q = 6 and 4 sin Q + 3 cos P = 1, then ∠R is equal to ______.
The value of cos (270° + θ) cos (90° – θ) – sin (270° – θ) cos θ is ______.
The value of the expression sin6 θ + cos6 θ + 3 sin2 θ . cos2 θ is ______.
The value of tan 315° cot(– 405°) + cot 495° tan (– 585°).
The value of `2 sin^2 π/6 + "cosec"^2 (7π)/6 cos^2 π/3` is ______.
cos2 5° + cos2 10° + cos2 15° + .... + cos2 85° + cos2 90° is equal to ______.
sin (270° – θ) sin (90° – θ) – cos ( 270° – θ) cos (90° + θ) is ______.
