# Prove ∠XOM = θ

## Homework Statement

[/B]Q7 part a on one of the attached pictures

2. Homework Equations

Trigonometric identities

## The Attempt at a Solution

See attached pages

Please help me I've spent onwards of 4 hours trying to figure this out and I can't get anywhere at all

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I have doubt about the problem. What is the guarantee that the line segment ZM passes through O the center of the circle. Does it mean that this is also given as input data.

## Homework Statement

[/B]Q7 part a on one of the attached pictures

2. Homework Equations

Trigonometric identities

## The Attempt at a Solution

See attached pages

Please help me I've spent onwards of 4 hours trying to figure this out and I can't get anywhere at all
∠XOY=2θ(angle at center twice angle at circumference)
XO=OY=raidius
∠XOM=∠YOM(angle at same segment)
2∠XOM=2θ
∠XOM=θ

I have doubt about the problem. What is the guarantee that the line segment ZM passes through O the center of the circle. Does it mean that this is also given as input data.

All I know is what I was given in the question. But I agree the question is presumptuous; it's assuming h≥r

∠XOY=2θ(angle at center twice angle at circumference)
XO=OY=raidius
∠XOM=∠YOM(angle at same segment)
2∠XOM=2θ
∠XOM=θ

How do you know angle XOY is double angle XZY? Is it just a rule that I don't know about?

How do you know angle XOY is double angle XZY? Is it just a rule that I don't know about?

Thank you so much that has helped heaps

The rest of the problem is quite simple. We need to solve the problem as mentioned in the diagram h = ZO +OM = r + rcos θ or
h/r = cos θ + 1. You can find dh/dθ and put r = 3 and θ = π/6 to complete the answer. The diiagram given in the book is a special case of the inscribed triangle where ZM passes through O.