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प्रश्न
Given: sin θ = `p/q`.
Find cos θ + sin θ in terms of p and q.
उत्तर
Consider the diagram below :
sin θ = `p/q`
i.e.`"perpendicular"/"hypotenuse" = p/h`
Therefore if length of perpendicular = px,
length of hypotenuse = qx
Since
hypotenuse2 = base2 + perpendicular2 ...[Using Pythagoras Theorem]
(qx)2 = base2 + (px)2
q2x2 = p2x2 + base2
q2x2 - p2x2 = base2
(q2 – p2)x2 = base2
∴ base = `sqrt(("q"^2 – "p"^"2")x^2)`
∴ base = `xsqrt("q"^2 - "p"^2) = "base"`
Now
cos θ = `"base"/"hypotenuse" = (xsqrt(q^2 – p^2))/(qx)`
Therefore, cosθ + sinθ
= `(xsqrt(q^2 – p^2))/(qx) + p/q`
= `(sqrt(q^2 – p^2))/(q) + p/q`
= `(p + sqrt(q^2 – p^2))/q`
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