an ocean grapher measured a set of waves during a storm and modelled the vertical displacement of waves using h(t)=0.6cos2t+0.8sint. Determine the vertical displacement of the wave when the velocity is 0m/s
the velocity is h'(t) = -1.2 sin2t+0.8cost
so, when is that zero?
t = π/2, sin^-1(1/3), π-sin^-1(1/3)
h(π/2) = -0.6 + 0.8 = 0.2
h(sin^-1(1/3)) = 0.6√8 + 0.8 * 1/3
h(π-sin^-1(1/3)) = -0.6√8 + 0.8 * 1/3
how do you know the function is at zero at sin^-1(1/3) and π-sin^-1(1/3) algebraically?
huh? You've forgotten your trig? Bad sign...
-1.2 sin2t+0.8cost = 0
-2.4 sint cost + 0.8 cost = 0
-0.8 cost (3sint - 1) = 0
cost = 0
sint = 1/3
To determine the vertical displacement of the wave when the velocity is 0 m/s, we need to find the values of t for which the velocity is 0 m/s. In this case, the velocity of the wave can be found by taking the derivative of the given displacement function, h(t).
Given: h(t) = 0.6cos(2t) + 0.8sin(t)
To find the velocity, v(t), we take the derivative of h(t) with respect to t:
v(t) = dh(t)/dt
Using basic differentiation rules, we differentiate each term of the equation:
v(t) = -1.2sin(2t) + 0.8cos(t)
Now, we need to find the values of t for which v(t) equals 0 m/s:
-1.2sin(2t) + 0.8cos(t) = 0
To solve this equation, we can rearrange it to isolate the trigonometric functions:
-1.2sin(2t) = -0.8cos(t)
Dividing both sides by -1.2 and using the identity sin(-x) = -sin(x), we get:
sin(2t) = (0.8/1.2)cos(t)
sin(2t) = (2/3)cos(t)
Now, we can use the trigonometric identity sin^2(x) + cos^2(x) = 1 to rewrite the equation:
(1 - cos^2(2t)) = (2/3)cos(t)
Expanding the square and rearranging the terms:
cos^2(2t) + (2/3)cos(t) - 1 = 0
This equation is a quadratic equation in terms of cos(t). We can solve the quadratic equation using factoring, completing the square, or applying the quadratic formula.
Once we solve the equation and find the values of cos(t), we can substitute them back into h(t) = 0.6cos(2t) + 0.8sin(t) to determine the corresponding vertical displacement of the wave when the velocity is 0 m/s.