Question

evaluate csc(3π/4)


please show work

Answer 1

Answer:

csc(3π/4) = √2

Step-by-step 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

Answer 2

Answer:

$\sqrt{2}$

Step-by-step explanation:

Using the trigonometric identity

• csc x = $\frac{1}{sinx}$

sin( $\frac{3\pi }{4}$) = sin ($\frac{\pi }{4}$) = $\frac{1}{\sqrt{2} }$, hence

csc ($\frac{3\pi }{4}$) = $\frac{1}{\frac{1}{\sqrt{2} } }$ = $\sqrt{2}$