Question

If tan θ =`15/ 8 `, find 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 tan θ = `"𝑃𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟"/" 𝐵𝑎𝑠𝑒"=(AB)/(BC) = 15/8`

So, if BC = 8k, then AB = 15k where k is positive number.
Now, using Pythagoras theorem, we have:
`AC^2 = AB^2 + BC^2 = (15K)^2 + (8K)^2`
`⟹ AC^2 = 225K^2 + 64^2 = 289^2`
⟹ AC = 17k
Now, finding the other T-ratios using their definitions, we get:

Sin 𝜃 = `(AB)/(AC) = (15 K)/(17K) = 15/17`

Cos θ = `(BC)/(AC) = (8K)/(17K)  = 8/17`

∴ cot 𝜃 = `1/(tan θ ) = 8/15 , cosec θ = 1/(sin θ ) = 17/15 and sec θ  = 1/(cos θ ) = 17/8`