what are the amplitude and period of the sinusoid y=6cosx+7sinx?
We could write 6cosx + 7sinx as a single
sine function of the form asin(x + A)
asin(x+A) = asinxcosA + acosxsinA = 7sinx + 6cosx
let x=0
---> asinA = 6 or sinA = 6/a
let x=π/2
---> acosA = 7 or cosA = 7/a
but sin^2 A + cos^2 B = 1
36/a^2 + 49/a^2 = 1
a^2 = 85
a =√85
so the amplitude is √85
and the period is 2π, (A is simply a phase shift)
To find the amplitude and period of the sinusoid y = 6cos(x) + 7sin(x), we need to understand the general form of a sinusoidal function.
A sinusoidal function can be written as y = A * cos(Bx + C) + D * sin(Bx + C), where A is the amplitude, B determines the period (B = 2π / period), C indicates a phase shift, and D is another amplitude.
In this case, y = 6cos(x) + 7sin(x) can be rewritten as y = (6 * cos(x)) + (7 * sin(x)), which matches the general form.
From the equation, we can see that A = 6 and D = 7. Therefore, the amplitude is the larger value between A and D, which is 7.
To determine the period, we need to find the value of B from the equation. Since B = 2π / period, we can rearrange the equation to solve for the period.
B = 1 in this case because we can see that x appears only once in both cos(x) and sin(x), indicating that the coefficient of x is 1.
Thus, the period is T = 2π / B = 2π / 1 = 2π.
In summary, the amplitude of the sinusoid y = 6cos(x) + 7sin(x) is 7, and the period is 2π.