Posted by Shin on Monday, May 27, 2013 at 11:51am.
Given that sin(2θ)=2/3, the value of
sin^6θ+cos^6θ can be written as a/b with a and b as coprime positive integers. Find a+b.
Trigonometry - Reiny, Monday, May 27, 2013 at 12:33pm
sin^6 θ+cos^6 θ
= (sin^2 Ø)^3 + (cos^2 Ø)^3 ---- the sum of cubes
= (sin^2 Ø + cos^2 Ø)( (sin^2 Ø)^2 - (sin^2 Ø)(cos^2 Ø) + (cos^2 Ø)^2 )
= (1)( (sin^2 Ø)^2 - (sin^2 Ø)(cos^2 Ø) + (cos^2 Ø)^2 )
now, if sin 2Ø = 1/3
2sinØcosØ = 2/3
sinØcosØ = 1/3
(sin^2 Ø)^2 + (cos^2 Ø)^2
= (sin^2 Ø + cos^2 Ø)^2 - 2(sin^2 Ø)(cos^2 Ø)
= (1 - 2(sinØcosØ)^2)
= 1 - 2(1/9) = 7/9
then (1)( (sin^2 Ø)^2 - (sin^2 Ø)(cos^2 Ø) + (cos^2 Ø)^2 )
= (1)(7/9 - 1/9)
so for you a+b stuff, a=2 and b=3
and a+b= 5
Answer This Question
More Related Questions
- Math - Calculus - The identity below is significant because it relates 3 ...
- trigonometry - It is given that sin(θ)=15√. In addition, a and b are ...
- Trignometry - Let θ be the angle between the x-axis and the line connecting...
- trigonometry - if cosθ=2/3 and tanθ<0 find the exact value sin(...
- maths - It is given that sin(θ)=1/√15. In addition, a and b are ...
- Math Question - Given tan θ = -8/5 and sin θ < 0, find sin θ, ...
- Trigonometry - Find the remaining trigonometric ratios of θ if sin(θ) ...
- Trig - Given that csc θ = -4 and tan θ > 0, find the exact value of...
- Calculus-antiderivative - Find f. f ''(θ) = sin θ + cos θ, f(0...
- Trigonometry - Could someone please help me with the following q's I'm having ...