If cos ( θ ) = 5/9 and θ is in the 1st quadrant, find sin ( θ )

Question

If cos ( θ ) = 5/9 and θ is in the 1st quadrant, find sin ( θ )

sin(0)=

Answer 1

Using inverse trigonometry,

cos(θ) = 5/9 ⇒ θ = cos^-1(5/9) ≈ 56.25 degrees

sin(θ) = sin(cos^-1(5/9)) ≈ sin(56.25) ≈ 0.8315

A more clever way would be to invoke the identity sin²(θ) + cos²(θ) = 1 such that

sin(θ) = sqrt(1 - cos²(θ)) = sqrt(1 - (5/9)²) = sqrt(1 - 25/81) = sqrt(56/81) = 2/9 * sqrt(14)

Note that we took the positive root because we are in quadrant 1.