How to find the angles of a triangle in a semicircle?

[Helly123]

Homework Statement

Homework Equations

d(y)/d(x) --> max area
area of triangle = 1/2 . base . height

The Attempt at a Solution

for number (2) [/B]

x^2 + y^2 = r^2 --> circle equation
base = 2R, height = y
Area = 1/2 . 2R . y
area = 1/2 . 4. √ (r^2 - x^2)

area now is half of max = 2, so :
2 = 1/2 . 4. √ (r^2 - x^2)
2 = 2 √(4-x^2)
1 = √ (4-x^2)
1 = 4 - x^2
x = √3
now y = 1
AP = 2 - √3
I have to find w (angle of ACB)
PC = 1
how to find w ?

 

[Doc Al]

Consider the triangle AOC.

 

[Helly123]

Consider the triangle AOC.

OC^2 = AC^2 + AO^2 -2AC.AO.cosw
4 = 4 + (8-4v3)^2 -8(8-4v3)cosw
Cos w = (8-4v3)/8
Cos w = 1-1/2v3

But the answer is in degrees, and not allowed using calculator.. And I don't think my answer is right.

Note: v3 = root 3

 

[Doc Al]

Consider the triangle AOC. You can find the angle at O by studying the right triangle CPO.

 

[Helly123]

I don't get it.. Can anyone help??

 

[ehild]

I don't get it.. Can anyone help??

CPO is a right triangle. You know the hypotenuse, it is one radius, length 2. You calculated PC, it is 1. What is the blue angle then?

 

[Helly123]

View attachment 205500
CPO is a right triangle. You know the hypotenuse, it is one radius, length 2. You calculated PC, it is 1. What is the blue angle then?

The blue angle is 60degrees.

 

[ehild]

The blue angle is 60degrees.

No. Imagine half of an equilateral triangle, rotated

 

[Helly123]

No. Imagine half of an equilateral triangle, rotated

Oh.. 30degrees ...

 

[Helly123]

No. Imagine half of an equilateral triangle, rotated

So, i get it. One triangle = 180degrees. I get 30, left 150. CAO = ACO. So CAO = 75 degrees?

 

[Helly123]

No. Imagine half of an equilateral triangle, rotated

But with theory, it should be sin w/ 2 = sin tetha / AC
I get sin w = 1/AC
While AC = 8-4root3
How can i get w with this method?

 

[ehild]

So, i get it. One triangle = 180degrees. I get 30, left 150. CAO = ACO. So CAO = 75 degrees?

Yes, it is correct.

 

[ehild]

But with theory, it should be sin w/ 2 = sin tetha / AC

What is theta, and what "theory" is that?

I get sin w = 1/AC
While AC = 8-4root3
How can i get w with this method?

Check with a calculator. Otherwise it is complicated.