How is the area under the arch of y=sinx related to the unit circle?

[Trav44]

Hi guys I'm new to the forum but with year 12 maths coming up next year I assume I'll be coming here a lot.

This question is part of my end of year investigation.

I have scanned the questions and attached them as jpegs.

I was thinking something along the lines of using the triangle to determine the area? Or if the fact that this curve is related to half of a unit circle had something to do with it?

Any help would be much appreciated :)

Thank you, Travis

 

[SteamKing]

You've got to show an attempt at solving the problem in order to get some help.

 

[CompuChip]

Hi Trav, welcome to the forums. I do hope you'll be coming here - and learning - a lot.
As SteamKing said, please show us your attempt or problem. The questions are leading you in a specific direction. As you've noticed, there is a pretty big hint in the picture in the form of the auxiliary lines that have been drawn, creating for example a triangle. If you show us how far you got with that, we may be able to help you complete it.

 

[Trav44]

Sorry I was unaware that an attempt must be shown, I will be sure to include them in every question from here on out :)
My attempts for each question as follows are:
1. The area of the outer rectangle is pi as pi x 1 = pi
The area for the inner triangle is 1/2bh so 1/2 x pi x 1= 1/2pi or pi/2
this shows that the area under the sine arch must fit the range of pi./2 < x < pi

2.
Triangle area= 1/2 x b x h
1/2 x pi/4 x sin(pi/4)= 0.2776

Trapezium area = (a + b)/2 x h
(sin (pi/4) + sin(pi/2))/2 x pi/4=0.6704

total area = 0.2776x2 + 0.6704x2= 1.896

which is the correct area

I get that sin of (pi/4 or pi/2) is the height at those given points but could someone please give me an explanation as to why this is?

I have found the area for both triangles and trapeziums and multiplied them by 2 to give me the overall area and this is right how ever I cannot make the connection between this and the equation pi/4(1+(root 2))3. I can use the formula to add up all the areas and I get an answer close to 2 which seems logical as the area under a sine curve is meant to equal 2.
I do not understand how this formula has been derived though?
And for pi to be split into increments of 0.1 I thought the start of the equation should be 1/31.4 as this is pi/0.1?
1/20 would give increments close to 0.15 wouldn't it?