The sum of the three smallest positive values of theta such that 4(cos*theta)(sin*theta) = 1 is k*pi. Find k.
Help!!!!!
To solve the equation 4(cosθ)(sinθ) = 1, we need to find the values of theta that satisfy the equation. Let's break it down step by step:
1. Start with the equation: 4(cosθ)(sinθ) = 1
2. Divide both sides by 4: (cosθ)(sinθ) = 1/4
3. Recognize that (cosθ)(sinθ) can be rewritten as (1/2)(2cosθ)(2sinθ): (1/2)(2cosθ)(2sinθ) = 1/4
4. Use the double-angle formula for sine: sin2θ = 2sinθcosθ. This can be rearranged as: (2sinθcosθ) = sin2θ/2
5. Replace (2sinθcosθ) in the equation: (1/2)(sin2θ/2) = 1/4
6. Simplify: sin2θ/4 = 1/4
7. Multiply both sides of the equation by 4: sin2θ = 1
8. Take the square root of both sides: sinθ = ±√(1)
9. Since we want only positive angles, we take the positive square root: sinθ = 1
10. Find the value of theta when sinθ = 1: θ = π/2
The value of θ that satisfies the equation is π/2. However, we need to find the sum of the three smallest positive values of theta.
The three smallest positive solutions for θ are: π/2, π/2 + π, and π/2 + 2π.
To find the sum, add up the three values: π/2 + (π/2 + π) + (π/2 + 2π)
Simplifying this expression, we get: 3π
Therefore, the value of k is 3. The sum of the three smallest positive values of theta is 3π.
I hope this explanation helps!
It is saying that 39/12 which is 13/4, is wrong.
2t = π/6 or 5π/6 or 13π/6
t = π/12 or 5π/12 or 13π/12
sum = 19π/12
Thanks steve!!!! It is correct! Also thanks Damon for giving this an attempt!
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cos T * sin T = 1/4
so
2 cos T * sin T = 1/2
so
sin 2 T = 1/2
thus
2 T = 30 deg or pi/6 radians
T = pi/12 radians or 15 deg
now cos * sin must be +
so T in quadrant 1 or quadrant 3
so actually T = pi/12 or T = 13pi/12 which is 195 deg
pi/12 + 13 pi/12 + (2pi+pi/12)
= (1/12 +13/12 +24/12 + 1/12)pi
= (39/12)pi
check my arithmetic !