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

1.	If cos θ = \(\frac{4}{5}\), find all other trigonometric ratios of angle θ.
Answer» 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 = \(\frac{4}{3}\)

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

Answer»

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