Simple geometry problem: angle PQR = θ and angle QPR = 2θ. Prove that cos θ = p/(2q)

Enceladus

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Hi - could someone help me with this, please:

In a triangle PQR, angle PQR = θ and angle QPR = 2θ. Prove that cos θ = p/(2q).

This is meant to be solved without using the trig identity sin 2θ = 2 sin θ cos θ, but it is intended for the sine rule to be used. Using the sine rule, I have got as far as

p / sin 2θ = q / sin θ

but I can't get any further. Help much appreciated!
 
Hi - could someone help me with this, please:

In a triangle PQR, angle PQR = θ and angle QPR = 2θ. Prove that cos θ = p/(2q).

This is meant to be solved without using the trig identity sin 2θ = 2 sin θ cos θ, but it is intended for the sine rule to be used. Using the sine rule, I have got as far as

p / sin 2θ = q / sin θ

but I can't get any further. Help much appreciated!
How do p and q relate to P, Q, and R?

Thank you!

Eliz.
 
Assuming that [imath]p[/imath] and [imath]q[/imath] are the lengths of the sides opposite to P and Q respectively, here is a hint:

Define point S on PQ so that PR = SR (i.e., triangle PRS is isosceles) , then consider the SQR triangle.
 
Hi - could someone help me with this, please:

In a triangle PQR, angle PQR = θ and angle QPR = 2θ. Prove that cos θ = p/(2q).

This is meant to be solved without using the trig identity sin 2θ = 2 sin θ cos θ, but it is intended for the sine rule to be used. Using the sine rule, I have got as far as

p / sin 2θ = q / sin θ

but I can't get any further. Help much appreciated!
You're almost there!

What is the identity for sin(2θ)?
 
Hi - could someone help me with this, please:

In a triangle PQR, angle PQR = θ and angle QPR = 2θ. Prove that cos θ = p/(2q).

This is meant to be solved without using the trig identity sin 2θ = 2 sin θ cos θ, but it is intended for the sine rule to be used. Using the sine rule, I have got as far as

p / sin 2θ = q / sin θ

but I can't get any further. Help much appreciated!
Clearly I failed to read the whole question.

But how do you know what is "meant" and "intended"? Can you quote the entire problem as given to you?

In effect, this problem derives the angle sum identity! Is the idea that that identity has not been taught yet, though the law of sines has, and someone is using this to prove the former? Anything you can say about the context may help.

But I have found a solution, which turns out to be equivalent to what @blamocur must have in mind. I did it by constructing the perpendicular bisector of QR ...
 
Clearly I failed to read the whole question.

But how do you know what is "meant" and "intended"? Can you quote the entire problem as given to you?

In effect, this problem derives the angle sum identity! Is the idea that that identity has not been taught yet, though the law of sines has, and someone is using this to prove the former? Anything you can say about the context may help.

But I have found a solution, which turns out to be equivalent to what @blamocur must have in mind. I did it by constructing the perpendicular bisector of QR ...
Yes, the idea is that the double angle identity has not yet been taught but the sine rule has. The problem in its entirety is:

In a triangle PQR, angle PQR = θ and angle QPR = 2θ. Prove that cos θ = p/(2q).

It is in a section on the sine rule; neither the cosine rule nor any trigonometric identities have been introduced yet.
 
Yes, the idea is that the double angle identity has not yet been taught but the sine rule has. The problem in its entirety is:

In a triangle PQR, angle PQR = θ and angle QPR = 2θ. Prove that cos θ = p/(2q).

It is in a section on the sine rule; neither the cosine rule nor any trigonometric identities have been introduced yet.
As I mentioned over at MHF, what formulas are you allowed to use to find cosine? Until we know that we can't set up an equation to solve for it! I think I have the same solution that blamocur and Dr.Peterson came up with but until we know what you are allowed to use, we can't get to the final answer!

-Dan
 
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