Explanation:
radian mode means that the argument of cosine has to be seen in radians (arc length units instead of degrees).
1 radian = the angle in a circle, so that the circle arc is exactly as long as the radius (1 unit).
the circumference of a circle is
2×pi×r
in the norm circle r = 1, so we get
2×pi = 6.283185307...
and the radius fits 6.283185307... times into the full circle circumference.
that means
1 radian corresponds to 360/6.283185307... = 57.29577951...°
in other words, every 2pi (6.283185307...) we went around the whole circle, and sine(2pi) is 0 and cosine(2pi) is 1.
any other argument for cosine is calculated in relation to these 2pi.
for example in (b) we get 4×3 = 12 as argument for cosine. and that means we are almost 2 times around (4pi) the whole circle. but only almost, as 4pi = 12.56637061...
I hope that helps you understand the background better.
for this example we just have to use the given values in the formula and calculate (but the cosine in radian mode - not in degree mode).
(a) t = 0 (right before the motion starts)
d = 0.4×e^(-0.2t) × cos(4t) =
= 0.4×e^(-0.2×0) × cos(4×0) =
= 0.4×e⁰ × cos(0) = 0.4×1×1 =
= 0.4 in
that means that somebody pulled the weight 0.4 inches off the equilibrium (where everything is just resting) to start the swinging motion.
(b) t = 3 (seconds into the motion)
d = 0.4×e^(-0.2×3) × cos(4×3) =
= 0.4×e^(-0.6) × cos(12) =
= 0.4×0.548811636... × 0.843853959... =
= 0.185246749... in