Question

If cos θ  = `7/25` find the value 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 cos 𝜃 =  `"𝐵𝑎𝑠𝑒" /"ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠" = (BC)/(AC) = 7/25`

So, if BC = 7k, then AC = 25k, were k is a positive number.
Now, using Pythagoras theorem, we have:

`AC^2= AB^2 + BC^2`

`⟹ AB^2 = AC^2 − BC^2 = (25K)^2 − (7K)^2`

`⟹ AB^2 = 625K^2 − 49K^2 = 576^2`

⟹ AB = 24k
Now, finding the trigonometric ratios using their definitions, we get:

Sin 𝜃 =` (AB)/(AC) = (24K)/(25K) = 24/25`

Sin 𝜃 =`(AB)/(BC) = (24K)/(7K) = 24/7`

∴ cot 𝜃 = `1/ (tan θ) = 7/24 , cosec  θ =  1/(sin θ) = 25/24 and sec θ = 1/ (cos θ) = 25/7`