Solve the trigonometric equation sin^17 + cos^16 = 1

[Math Tutor]

Solve the trigonometric equation:
[tex]sin^{17}x + cos^{16}x = 1[/tex]

[Guest]

Solve for x, sin(x)=0 and sin(x)=1. Then *U* them.

YS.

[Math Tutor]

How do you prove that there are no other solutions?

[Guest]

Who doesn't know it?
For all non (pi/2)n we have |sin(x)|<1, |cos(x)|<1 and (sin(x))^17 < (sin(x))^2 and
(cos(x))^16 < (cos(x))^2. Q.E.D.

YS. (Is it enough? This problem could be generalized taking into account the powers used)

[Guest]

Mrs. Teacher, i also offer u a question based upon your question (and u can resort to all teachers u know).
If we use the notation sin(x)=t in sin^17 + cos^16 = 1 then we get:
t^2(t-1)(t^14 + 2t^13 + 2t^12 - 6t^11 - 6t^10 + 22t^9 + 22t^8 - 34t^7 - 34t^6 + 36t^5 + 36t^4 - 20t^3 - 20t^2 + 8t + 8)=0

Now let's cut only the big polynomial:
t^14 + 2t^13 + 2t^12 - 6t^11 - 6t^10 + 22t^9 + 22t^8 - 34t^7 - 34t^6 + 36t^5 + 36t^4 - 20t^3 - 20t^2 + 8t + 8=0 and see how u show that it has no real solution in [-1,1]. Prove it without taking into consideration initial equation and its proof, or any
calculator, or just giving some values to t in [-1,1]. A good proof is required.


C.(i can wait a month for it)

[amit28it]

t^2(t-1)(t^14 + 2t^13 + 2t^12 - 6t^11 - 6t^10 + 22t^9 + 22t^8 - 34t^7 - 34t^6 + 36t^5 + 36t^4 - 20t^3 - 20t^2 + 8t + 8)=0
is enlarge form of it.

[Guest]

This is true but it not the expected solution.


Y0uRShAD0W

[Guest]

There is a simple solution.

sin(x)¹⁷+cos(x)¹⁶=1
sin(x)¹⁷+cos(x)¹⁶-1=0

Remember that sin(x)²+cos(x)²=1. So
(sin(x)⁸)²+(cos(x)⁸)²=1, thus
sin(x)¹⁶+cos(x)¹⁶=1 and
cos(x)¹⁶=1-sin(x)¹⁶

Substituting cos(x)¹⁶ with 1-sin(x)^16 into sin(x)¹⁷+cos(x)¹⁶-1=0 gives
sin(x)¹⁷+1-sin(x)¹⁶-1=0
sin(x)¹⁷-sin(x)¹⁶=0
Factor:
sin(x)¹⁶ (sin(x)-1)=0

sin(x)¹⁶=0 when [tex]x={0, \pi }[/tex]
sin(x)-1=0 when [tex]x=\pi /2[/tex]

Also, try graphing the solution. See the output at Wolfram Alpha:
http://www.wolframalpha.com/input/?i=si ... +x+%3C+2pi (Solutions near bottom.)

[Guest]

Too long solution. Just take a look to my inequality way!