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

Enceladus

New member
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!

stapel

Super Moderator
Staff member
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.

blamocur

Senior Member
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.

Dr.Peterson

Elite Member
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)?

Enceladus

New member
You're almost there!

What is the identity for sin(2θ)?
It needs to be proved without using the sin(2θ) identity, as stated in the problem.

The Highlander

Full Member
You're almost there!

What is the identity for sin(2θ)?
Indeed!
But...

This is meant to be solved without using the trig identity sin 2θ = 2 sin θ cos θ

though I don't see how it can be proved without introducing something the relates the Sine to the Cosine!

Dr.Peterson

Elite Member
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 ...