# The motion of an oscillator is described by the equation x(t)=sin t+sin2t. What is x if cos t=-0.25 and t is in the interval (180, 270)?

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First of all, before calculate sin t, we must establish to what quadrant belongs. Due to the facts from hypothesis, t is in the interval (pi, 3pi/2), so the angle t belongs to the third quadrant, where the value of the function sine is negative.

cos a = -.25 = -1/4

sin a = sqrt[1- (-1/4) (from the fundamental formula of trigonometry,where (sin a)^2 + (cosa)^2 = 1).

sin a = -sqrt(15)/4

To determine x, first we have to calculate sin 2t.

We'll apply the formula for the double angle:

sin 2a = sin (a+a)=sina*cosa + sina*cosa=2sina*cosa

We'll substitute 2a by 2t and we'll re-write the equation x(t).

x(t) = sin t + 2sint*cos t

e= -sqrt(15)/4 + 2*(1/4)*sqrt(15)/4

We'll calculate the LCD of the ratios:

LCD = 16

We'll factorize by sqrt(15)/4:

x(t) = [sqrt(15)/4](-1 + 2/4)

x(t) = [sqrt(15)/4](-1 + 1/2)

**x(t) = [-sqrt(15)/8]**

The given oscillation function is x(t) = sint+sin2t

To find x if cost = -0.25 if tis in (180 , 270).

When cost = -0.25, sint = - sqrt(1-cos^2t) = -sqrt(1-0.25^2)

sint = -0.968245836...(1).

sin2t = 2sint*cost.

sin2t = 2{-sqrt(1-.25)^2}{-.25}

sin2t = 0.484122918....(2).

We substititute the obtained value of sint and sin2t in thye given equation, x(t) = sint+sin2t and get:

x(t) = -0.968245836 + 0.484122918.

x(t) = -0.484122918.

Therefore , when cost = -0.25 fot in (180,270), the value x = sint+sin2t = -0.484122918.