Please see the attachment and have a look at the diagram for this question.
This is a question to prove double angle formulae using the diagram of a semi-circle of radius 1 unit.
As you can see, it says as follows:
In the diagram, the semi-circle has radius 1 unit, and angle PAB = theta.
Angle APO = theta {Triangle AOP is isosceles}
Angle PON = 2*theta {exterior angle of a triangle}
a) Find in terms of theta, the lengths of:
i) [OM]
sin (theta)
ii) [AM]
cos (theta)
iii) [ON]
cos (2*theta)
iv) [PN]
sin (2*theta)
So far so easy. But I have difficulty with the following question:
b) Use the triangle ANP and the lengths in a) to show that:
i) cos (theta) = sin (2*theta)/2*sin (theta)
From a) above,
This is a question to prove double angle formulae using the diagram of a semi-circle of radius 1 unit.
As you can see, it says as follows:
In the diagram, the semi-circle has radius 1 unit, and angle PAB = theta.
Angle APO = theta {Triangle AOP is isosceles}
Angle PON = 2*theta {exterior angle of a triangle}
a) Find in terms of theta, the lengths of:
i) [OM]
sin (theta)
ii) [AM]
cos (theta)
iii) [ON]
cos (2*theta)
iv) [PN]
sin (2*theta)
So far so easy. But I have difficulty with the following question:
b) Use the triangle ANP and the lengths in a) to show that:
i) cos (theta) = sin (2*theta)/2*sin (theta)
From a) above,
sin (2*theta) = [PN], which is the opposite side from the angle PAB (theta) in the triangle ANP.
But I cannot identify 2*sin (theta), which is 2*[OM], in the triangle ANP.
Why double [OM]?
Why do we need to divide [PN] by double [OM] to get cos (theta)???
I would much appreciate it if someone can help me to make sense of this equation above from this diagram.
Thank you.
But I cannot identify 2*sin (theta), which is 2*[OM], in the triangle ANP.
Why double [OM]?
Why do we need to divide [PN] by double [OM] to get cos (theta)???
I would much appreciate it if someone can help me to make sense of this equation above from this diagram.
Thank you.