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Question
If cos θ = `7/25` find the value of all T-ratios of θ .
Solution
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`
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