Suppose θ is an angle strictly between 0 and π/2 such that sin5θ=(sin^5)θ. The number tan2θ can be uniquely written as a√b, where a and b are positive integers, and b is not divisible by the square of a prime. What is the value of a+b?
well, let's see. Leaving out all the x's,
sin(5x) = sin^5 + 5sin cos^4 - 10 sin^3 cos^2
= sin^5 + 5sin^5 - 10sin^3 + 5sin - 10sin^3 + 10sin^5
= 16sin^5 - 20sin^3 + 5sin
so, if sin(5x) = sin^5(x)
3sin^5 - 4sin^3 + sin = 0
(3sin-1)(sin-1) = 0
sin(x) = 1/3 or 0
x=0 is out, since it's not in (0,π/2)
So, sin(x) = 1/3
cos(x) = √8/3
sin(2x) = 2 sinx cosx = 2√8/9
cos(2x) = 1 - 2sin^2(x) = 7/9
tan(2x) = 2√8/7
Hmmm. I don't get a√b
Better double check my algebra.
the answer is 2*sqrt(3) or 3*sqrt(2) or 2*sqrt(2) or 3*sqrt(3).....try them.....
i still dun get it :(
copying from above
sin(5x) = = 16sin^5 - 20sin^3 + 5sin
sin(5x) = sin^5(x)
3sin^5 - 4sin^3 + sin = 0
(3sin^2 -1)(sin^2 -1) = 0
sin^2(x) = 1/3
tan x = 1/√2
tan 2x = 2√2
To find the value of a and b, we need to simplify the expression for tan(2θ) using the given information.
First, let's recall the double-angle formula for tangent:
tan(2θ) = (2tan(θ))/(1 - tan^2(θ))
Since we are given that θ is an angle strictly between 0 and π/2 and sin(5θ) = (sin(θ))^5, we can rewrite tan(θ) in terms of sin and cos.
Using the identity sin^2(θ) + cos^2(θ) = 1, we have:
sin(θ) = √(1 - cos^2(θ))
Now, let's use the given expression sin(5θ) = (sin(θ))^5 to simplify the equation:
(sin(θ))^5 = (sin(θ))^5
Using the expression sin(θ) = √(1 - cos^2(θ)):
(√(1 - cos^2(θ)))^5 = (√(1 - cos^2(θ)))^5
Expanding both sides of the equation:
1 - cos^2(θ) = 1 - cos^2(θ)
Since both sides of the equation are equal, we can conclude that cos^2(θ) must be equal to 0.
Therefore, cos(θ) = 0.
Now, we can substitute this value for cos(θ) into the double-angle formula for tangent:
tan(2θ) = (2tan(θ))/(1 - tan^2(θ))
tan(2θ) = (2 * (sin(θ)/cos(θ)))/(1 - (sin(θ)/cos(θ))^2)
Since cos(θ) = 0, the denominator becomes 1 - 0^2 = 1.
tan(2θ) = 2 * (sin(θ)/1)
tan(2θ) = 2sin(θ)
We know that sin(θ) = √(1 - cos^2(θ)), and since cos(θ) = 0, sin(θ) becomes:
sin(θ) = √(1 - 0^2) = √1 = 1
Substituting sin(θ) = 1 into the expression for tan(2θ):
tan(2θ) = 2 * sin(θ)
tan(2θ) = 2 * 1
tan(2θ) = 2
Therefore, a = 2, and b = 1.
The value of a+b is 2+1 = 3.