2.4. Use the cosine rule in $$ riangle \mathrm{POQ} $$ to calculate the length of PQ in terms of $$ \alpha $$ and $$ \beta $$.

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2.4. Use the cosine rule in $$ 	riangle \mathrm{POQ} $$ to calculate the length of PQ in terms of $$ \alpha $$ and $$ \beta $$.

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We start with triangle POQ and apply the cosine rule. In any triangle with sides a, b and with an included angle γ, the cosine rule says

(side opposite γ)² = a² + b² − 2ab·cos γ.

In our case, the side we need is PQ, and if we denote
  OP = 1 and OQ = 1            (for example, if P and Q lie on the unit circle)
and if we know that the angle at O is
  ∠POQ = α + β,
then the cosine rule gives

PQ² = OP² + OQ² − 2·OP·OQ·cos(α + β)
     = 1² + 1² − 2·1·1·cos(α + β)
     = 2 − 2cos(α + β).

Taking square roots of both sides we obtain

PQ = √[2 – 2cos(α + β)].
     = √[2(1 – cos(α + β))].

Recall now the half‐angle identity

1 − cosθ = 2sin²(θ/2).

Setting θ = α + β we have

1 − cos(α + β) = 2sin²[(α + β)/2].

Thus

PQ = √[2·2sin²((α + β)/2)]
     = √[4sin²((α + β)/2)]
     = 2|sin((α + β)/2)|.

For problems in which α and β (or their sum) are such that sin((α + β)/2) is non‐negative (which is often the case by convention), we may drop the absolute value and finally write

PQ = 2 sin((α + β)/2).

This is the required expression for PQ in terms of α and β.
The length of PQ is $$ 2 \sin\left(\frac{\alpha + \beta}{2}\right) $$.

2.4. Use the cosine rule in \( \triangle \mathrm{POQ} \) to calculate the length of PQ in terms of \( \alpha \) and \( \beta \).

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