Tan(pi/9)*tan(2pi/9)*tan(4pi/9)
Answers
Given: Tan(pi/9)*tan(2pi/9)*tan(4pi/9)
To find: The value of the given expression.
Solution:
- Now we have given the term tan(pi/9)*tan(2pi/9)*tan(4pi/9)
- Simplifying it, we get:
tan ( 20 ) x tan ( 40 ) x tan ( 80 )
- Now converting it to sin and cos, we get:
(sin 20 / cos 20 x sin 40 / cos 40 x sin 80 / cos 80)
- Multiplying by 2 in numerator and denominator, we get:
2 sin (20) x sin (40) x sin (80) / 2 cos 20 x cos 40 x cos 40
- Now we have the formula:
2 sin x sin y = cos(x-y) - cos(x+y) ..................(i)
2 cos x cos y = cos(x+y) + cos(x-y) ..................(ii)
- So applying it, we get:
(cos(20°-40°)-cos(20°+40°)) x sin80° / (cos(20°+40°)+cos(20°-40°)) x cos80°
( cos20° - cos60° ) x sin80° / ( cos60° + cos20° ) x cos80°
cos20° x sin80° - cos60° x sin80° / cos60° x cos80° + cos20° + cos80°
( cos20° x sin80° - sin80°/2 ) / ( cos80°/2 + cos20° + cos80° )
- Multiplying by 2 in numerator and denominator, we get:
( 2cos20° x sin80° - sin80° ) / ( cos80° + 2cos20° + cos80° )
- Now we have the formula:
2 sin x cos y = sin (x+y) - sin (x-y)
2 cos x cos y = cos (x+y) + cos (x-y)
( sin(20°+80°)-sin(20°-80°)-sin80° ) / ( cos80°+cos(20°+80°)+cos(20°-80°) )
(sin100°+sin60°-sin80°)/(cos80°+cos100°+cos60°)
( 2cos(100°+80°)/2sin(100°-80°)/2 +√3/2) / ( 2cos(100 +80°) / 2cos(100°-80°) /2+1/2 )
- Now simplifying this, we get:
( 2 cos90° sin10° + √3/2 ) / ( 2 cos90° cos10° + 1/2 )
(√3/2)/(1/2)
√3
Answer:
So the value of given term is √3.