What is the angle of the nth O in a decreasing rate?

[adjacent]

Homework Statement

Look at the image.All the O's are of equal length.O1=O2=O3 etc.Angles are subtended from two ends of An O to point P.As you see,O1's angle is larger than O2 ant )2's angle is larger then O3 and so on.
As you Go down the O's(i.e. O1,O2,O3 etc),the angle subtented by the O's Decrease and that too with a decreasing rate.
What would be the angle of nth O.

See the image Clearly please..

Homework Equations

a+d(n-1)
ar^(n-1)

The Attempt at a Solution

As the angles decrease with decreased rate,The Formulas are not usable.
What should I do?Angles are not even inversely proportional to O's.

 

[szynkasz]

You can use the cosine rule. Lengths of two "rays" can be calculated from Pythagorean theorem.

 

[adjacent]

You can use the cosine rule. Lengths of two "rays" can be calculated from Pythagorean theorem.

What could be the angle of nth O?
And I haven't studied Cosine rule yet.I am In Gr 9.

 

[szynkasz]

Rays are hypotenuses of right angles so you can use tangent function instead. The angle of nth O is a difference of angles of nth and n-1th triangle.

 

[adjacent]

Rays are hypotenuses of right angles so you can use tangent function instead. The angle of nth O is a difference of angles of nth and n-1th triangle.

I don't Understand

 

[szynkasz]

Let $PO$ be the segment from $P$ to the line of $O$'s Together with $a$ and $O_1$ it makes a right triangle so you can calculate $\tan\alpha_{\small O_1}$

 

[Strants]

Just to make sure everyone's on the same page: adjacent, do you know trigonometric functions (Sine, Cosine, Tangent)?

 

[adjacent]

Just to make sure everyone's on the same page: adjacent, do you know trigonometric functions (Sine, Cosine, Tangent)?

Yes.But
Only sineθ=Opposite/Hypotenuse
Cosineθ=Adjacent/Hypotenuse
a Tangentθ=Opposite/Adjacent
But that's for Right angle triangle only.Right?

Assume that we don't know any thing about the length of the lines.Only Length of O and Angles(on the side of P) are given.We are to calculate,for example,For O1,the angle is this much.For O2 The angle would be that much.Then For On what would be the angle?

 

[szynkasz]

$PO,2O,b$ also makes right triangle. So the angle for 2nd $O$ is $\alpha_{\small O_2}-\alpha_{\small O_1}$