Question

If 8 tanθ = 15, find (i) sinθ, (ii) cotθ, (iii) sin²θ - cot²θ

Answer 1

8tan θ = 15

⇒ tan θ = `(15)/(8) = "Perpendicular"/"Base"`

Hypotenuse
= `sqrt(("Perpendicular")^2 + ("Base")^2`
= `sqrt(15^2 + 8^2)`
= `sqrt(225 + 64)`
= `sqrt(289)`
= 17

(i) sin θ  = `"Perpendicular"/"Hypotenuse" = (15)/(17)`

cot θ  = `(1)/"tan θ " = (8)/(15)`

(iii) sin²θ  - cot²θ 
= (sin θ + cot θ)(sin θ  - cot θ)

= `(15/17 + 8/15)(15/17 - 8/15)`

= `((225 + 136)/225)((225 - 136)/225)`

= `(361/225)(89/255)`

= `(32129)/(65025)`.