Question

Given: sin θ = `p/q`.
Find cos θ + sin θ in terms of p and q.

Answer 1

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 = p²x^{2 +} 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`