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Question
If `sin theta = a/b` find sec θ + tan θ in terms of a and b.
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Solution
We know `sin theta = "𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒"/"ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒"`
Let x be the adjacent side
By applying Pythagoras theorem
𝐴𝐶2 = 𝐴𝐵2 + 𝐵𝐶2
b2 = a2 + x2
x2 = b2 − a2
`x = sqrt(b^2 - a^2)`
`sec theta = (AB)/(BC) = b/(sqrt(b^2 - a^2))`
`tan theta = (AB)/(BC) = a/(sqrt(b^2 - a^2))`
`sec theta + tan theta = b/(b^2 - a^2) + a/(sqrt(b^2 - a^2))`
`= (b + a)/(sqrt(b^2 - a^2)) = (b+ a)/sqrt((b + a)(b - a)) = (b + a)/sqrt(b + a) - 1/(sqrt(b - a)) = sqrt((b + a)/(b - a))`
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