If cos θ = 4/5, find all other trigonometric ratios of angle θ.

Question

If cos θ = \(\frac{4}{5}\), find all other trigonometric ratios of angle θ.

Answer 1

We have, 

cos θ = \(\frac{4}{5}\) 

And we know that, 

sin θ = √(1 – cos² θ)

⇒ sin θ = √(1 – (\(\frac{4}{5}\))²) 

= √(1 – (\(\frac{16}{25}\))) 

= √[\(\frac{(25 \,–\, 16)}{25}\)] 

= √(\(\frac{9}{25}\)) 

= \(\frac{3}{5}\)

∴ sin θ =\(\frac{3}{5}\)

Since, 

cosec θ = \(\frac{1}{sin\, θ }\)

= \(\frac{1}{(3/5)}\)

⇒ cosec θ = \(\frac{5}{3}\) 

And, sec θ = \(\frac{1}{cos θ }\)

= \(\frac{1}{(4/5)}\)

⇒ cosec θ = \(\frac{5}{4}\) 

Now, 

tan θ = \(\frac{sin θ}{cos θ }\)

= \(\frac{(3/5)}{(4/5)}\)

⇒ tan θ = \(\frac{3}{4}\) 

And, cot θ = \(\frac{1}{tan θ}\) 

= \(\frac{1}{(3/4)}\)

⇒ cot θ = \(\frac{4}{3}\)