Question

Find the value of the trigonometric functions for the following:
tan θ = −2, θ lies in the II quadrant

Answer 1

We know that sec²θ – tan²θ = 1

sec²θ – (– 2)² = 1

sec²θ – 4 = 1

sec²θ = 1 + 4 = 5

sec θ = `+-  sqrt(5)`

Since θ lies in the second quadrant sec θ is negative.

∴ sec θ = `- sqrt(5)`

cos θ = `1/sectheta = -1/sqrt(5)`

We know cos²θ + sin²θ = 1

`(- 1/sqrt(5))^2 + sin^2theta` = 1

`1/5 + sin^2theta` = 1

sin²θ = `1 - 1/5 = (5 - 1)/5`

sin²θ = `4/5`

sin θ = `+-  2/sqrt(5)`

Since θ lies in the second quadrant sin θ is positivee.

∴ sin θ = `2/sqrt(5)`

sin θ =  `2/sqrt(5)`, cosec  = `1/sintheta = sqrt(5)/2`

cos θ = `- 1/sqrt(5)`, sec θ = `1/costheta = - sqrt(5)`

tan θ =  – 2, cot θ = `1/tantheta = - 1/2`