Question

Evaluate csc(3π/4)


please show work

Answer 1

`√(2)`

Explanation:

Using the trigonometric identity

• csc x =
`(1)/(sinx)`

sin(
`(3\pi )/(4)`) = sin (
`(\pi )/(4)`) =
`(1)/(√(2) )`, hence

csc (
`(3\pi )/(4)`) =
`(1)/((1)/(√(2) ) )` =
`√(2)`

Answer 2

csc(3π/4) = √2

Explanation:

∵ csc(3π/4) = 1 ÷ sin(3π/4)

∵ ∠(3π/4) lies on the second quadrant

∴ sin(3π/4) is positive value (according to ASTC Rule)

* ASTC Rule ⇒ (All +ve in 1st quadrant , Sin +ve in 2nd , Tan +ve

in 3rd quadrant , Cos +ve in 4th quadrant)

∴ sin(3π/4) = sin(π - α) ⇒ where α is an acute angle

∴ 3π/4 = π - α ⇒ α = π - 3π/4 = π/4

∵ sin²x + cos²x = 1

∵ sin(π/4) = cos(π/4)

∴ 2sin²(π/4) = 1 ⇒ sin²(π/4) = 1/2 ⇒ sin(π/4) = √(1/2)

∴ sin (π/4) = 1/√2

∴ sin(3π/4) = 1/√2

∴ csc(3π/4) = 1 ÷ 1/√2 = 1 × √2/1 = √2