Wednesday

April 23, 2014

April 23, 2014

Posted by **Shin** on Monday, May 27, 2013 at 11:51am.

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 12:33pmsin^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

and

(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)

= 6/9

= 2/3

so for you a+b stuff, a=2 and b=3

and a+b= 5

**Related Questions**

Math - Calculus - The identity below is significant because it relates 3 ...

trigonometry - if cosθ=2/3 and tanθ<0 find the exact value sin(&#...

Algebra 2/ Trigonometry - The expression oos² θ/sin θ + sin θ is ...

Trigonometry - Which of the following is equivalent to 1 − cos 2 θ/...

Trig - Given that csc θ = -4 and tan θ > 0, find the exact value of...

Trigonometry - Find the remaining trigonometric ratios of θ if sin(θ) ...

Trigonometry - Could someone please help me with the following q's I'm having ...

Trig - Given that cos θ = 5/13 and θ is in QIV, find the exact value ...

Trigonometry - 1.Solve tan^2x + tan x – 1 = 0 for the principal value(s) to two ...

math - if cosθ=2/3 and tanθ<0 find the exact value sin(θ-5&#...