Question
If cos−1√p+cos−1√1−p+cos−1√1−q=3π4, then the value of q is
Answer: Option D
:
D
Let α=cos−1√p:β=cos−1√1−p
and γ=cos−1√1−q or cosα=√p:cosβ=√1−p
and cosγ=√1−q.
Therefore sinα=√1−p,sinβ=√p and sinγ=√q.
The given equation may be written as α+β+γ=3π4orα+β=3π4−γ or cos(α+β)=cos(3π4−γ)
⇒cosαcosβ−sinαsinβ = cos{π−(π4+γ)}=−cos(π4+γ)
⇒√p√1−p−√1−p√p=−(1√2√1−q−1√2.√q)
⇒0=√1−q−√q⇒1−q=q⇒q=12.
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:
D
Let α=cos−1√p:β=cos−1√1−p
and γ=cos−1√1−q or cosα=√p:cosβ=√1−p
and cosγ=√1−q.
Therefore sinα=√1−p,sinβ=√p and sinγ=√q.
The given equation may be written as α+β+γ=3π4orα+β=3π4−γ or cos(α+β)=cos(3π4−γ)
⇒cosαcosβ−sinαsinβ = cos{π−(π4+γ)}=−cos(π4+γ)
⇒√p√1−p−√1−p√p=−(1√2√1−q−1√2.√q)
⇒0=√1−q−√q⇒1−q=q⇒q=12.
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