Question

If cosec θ = `sqrt(10)` find all the values of all T-ratios of θ

Answer 1

Let us first draw a right ΔABC, right angled at B and ∠𝐶 = 𝜃
Now, we know that cosec θ = `"𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒"/" 𝑃𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟"= (AC)/(AB) = sqrt(10)/1`

So, if AC =` (sqrt(10))`𝑘, 𝑡ℎ𝑒𝑛 𝐴𝐵 = 𝑘 𝑖𝑠 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑛𝑢𝑚𝑏𝑒𝑟.
Now, by using Pythagoras theorem, we have:
`AC^2 = AB^2 + BC^2`
`⟹ BC^2 = AC^2 + BC^2`
`⟹ BC^2 = 9K^2`
⟹ BC = 3k
Now, finding the other T-ratios using their definitions, we get:

tan θ =`(AB)/(BC) = K/(3K) = 1/3`

cos θ = `(BC)/(AC) = (3K)/(sqrt(10 k)) = 3/(sqrt(10)`

∴ sin θ = `1/(cosec θ) = 1/(sqrt(10)) , cot θ = 1/ (tan θ ) = 3 and sec θ  1/(cos θ) = (sqrt(10))/3`