Maths parametric integration

The curve shown in the diagram has parametric equations:

x=acos3t
y=asint

-pi/6<=t<=pi/6

The curve meets the axes at point A(0,1/2), point B and point C(0,-1/2).

The straight lines shown are tangents to the curve at points A and C and meet the x-axes at point D.

a)Find in terms of 'a', the area of the finite region between the curve, the tangent at A and the x-axis.

I'm struggling with one thing in particular...how do we know what 't' is at point A since there's 2 possible options for 't' when x=0, t=pi/6 and t=-pi/6?

Edit: I'm not sure the questions actually possible.. how are you meant to integrate this parametrically?

Could you post a picture of the question with diagram?

The curve shown in the diagram has parametric equations:

x=acos3t
y=asint

-pi/6<=t<=pi/6

The curve meets the axes at point A(0,1/2), point B and point C(0,-1/2).

The straight lines shown are tangents to the curve at points A and C and meet the x-axes at point D.

a)Find in terms of 'a', the area of the finite region between the curve, the tangent at A and the x-axis.

I'm struggling with one thing in particular...how do we know what 't' is at point A since there's 2 possible options for 't' when x=0, t=pi/6 and t=-pi/6?

Edit: I'm not sure the questions actually possible.. how are you meant to integrate this parametrically?

a>0 so does that make the points unique?

How do we know that.Another question, how would we integrate this?

In the picture
x>0 so a>0.
It probably doesn't significantly affect the problem.

Assuming we know B & D,
area of triangle OAD - area under curve

Ah ok, I see why a>0 now , but how would we find the area under the curve? I'm getting to the stage where we have to integrate -3a^2*sint*sin3t which just looks wrong.

Its actually a reaonably simple integral.

Bit confused, so is -3a^2*sint*sin3t what we're meant to integrate or have I done something wrong?

I'm not sure what you've done, can you put a couple of lines in saying how you come up with it? Looks about right, but I've not worked it out fully.

dx/dt = -3asin(3t)

Therefore we can write our integral like this: asin(t) * -3asin(3t).

But I'm confused how we integrate this tbh.

tbh, its a fairly standard one
sin(u)sin(v) = (cos(u-v) - cos(u+v))/2
Easy to derive using the double angle formula.
Each of the cos terms can be easily integrated.

Wow lol. I've never seen that before. Thanks. Is there any other similar rules like this?

They're just the product / sum trig identities, useful when you want to integrate sin()sin() or cos()cos()
or even in more extreme cases sin()sin()sin(). They're just based on the cos(a+b) identities.
https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identities

They're just the product / sum trig identities, useful when you want to integrate sin()sin() or cos()cos()
or even in more extreme cases sin()sin()sin(). They're just based on the cos(a+b) identities.
https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identities