If cosx=(1/2 + (2^1/2)/2)^1/2 and sinx=-(1/2 - (2^1/2)/2)^1/2 0<x<2pi. It follows that 2x=kpi. The value of k is:
since cosx = √(1 + 1/√2)/2) cos2x = 1/√2, so 2x = π/4 or 7π/4
since cosx>0 and sinx<0 only in QIV,
3π/2 < x < 2π
that means 3π < 2x < 4π
Looks like k = 7/4
To find the value of k, we need to analyze the given equations for cos(x) and sin(x) and apply trigonometric identities.
First, let's simplify the given expressions for cos(x) and sin(x):
cos(x) = (1/2 + √2/2)^(1/2)
sin(x) = -(1/2 - √2/2)^(1/2)
We can recognize that the expressions inside the brackets (√2/2 and 1/2) correspond to the values of cos(π/4) and sin(π/4), respectively. To confirm this, we can represent these values as decimals:
cos(π/4) ≈ 0.707
sin(π/4) ≈ 0.707
Since cos(x) and sin(x) are positive in the given equations, we can infer that x lies in the first quadrant, where both trigonometric functions are positive.
Therefore, we can use the principle of symmetry in trigonometry. In the first quadrant, x and π/2 - x have the same values for cos(x) and sin(x). By applying this principle:
cos(x) = cos(π/2 - x) → (1/2 + √2/2)^(1/2) = cos(π/2 - x)
To find x, we need to solve for π/2 - x:
π/2 - x = arcsin[(1/2 + √2/2)^(1/2)]
Now, let's evaluate the arcsin of both sides:
π/2 - x = arcsin[(1/2 + √2/2)^(1/2)]
x = π/2 - arcsin[(1/2 + √2/2)^(1/2)]
Using a calculator, we can find the value of x ≈ 0.9553 radians.
Now, to find the value of k, we should consider the equation 2x = kπ.
Since we found x ≈ 0.9553, we can substitute it into the equation:
2(0.9553) = kπ
k = (2)(0.9553)/π ≈ 1.2103
Therefore, the value of k is approximately 1.2103.