15) The velocity of a particle moving along the x-axis is given by v(t)=sin(4t) at time t. If the particle is at x=6 at t=0, what is the position of he particle at =8 ?

A] the relationship between angles  ∠7 and  ∠8 are defined as vertically opposite angles, they are those angles that opposite to each other when two line cross.  they are vertical in the sense that they share the same vertex (the same corner angles) as opposed to the usual up-down meaning of vertical. 

b] the angles created by this intersection between c and d are special because they are equal to each other, to solve for y we shall have:
(5y-29)=(3y+19)
this is because they are vertical angles, hence solving the above we get:
5y-3y=19+29
2y=48
y=48/2
y=24°

c] to solve for  ∠7 and  ∠8 we proceed as follows;
angles in a straight line add up to 180:
therefore:
(5y-29)+∠7=180
substituting the value of y in the above and solving for  ∠7 we get:
(5×24-29)+∠7=180
91+∠7=180
∠7=180-91
∠7=89

but remember that  vertical angles are equal to each other, therefore:
∠7=∠8=89°