Given that sin =0.8387 and π/2<θ<π , use the Pythagorean identity to find cosθ.

Question

Given that sin =0.8387 and π/2<θ<π , use the Pythagorean identity to find cosθ.

Answer 1

sinx = 0.8387

Pythagorean Theorem is hypotenuse squared = sum of squares of the other two sides. on a unit circle the hypotenuse = 1, sinx and cos x are the other 2 sides.

sin^2x + cos^2x = 1^2

cos^2x = 1 - sin^2x

= 1 - (.8387)^2 = 1 - .7034

= 0.2966

cosx = square root of 0.2966 = -0.5446

(ignore the positive square root)

solution is -0.5446, round it off to the same number of decimal places as sinx = 0.8387. 4 decimal places

x = 180-57 = 123 degrees

if you use a calculator with an inverse trig function, it will say 57 degrees. That's the reference angle. But you want quadrant II, so subtract 57 from 180. 180-57= 123

123 times pi/180 = 0.6833 pi radians

= 2.1467 radians

cos(123) = -0.5446

check your answer by plugging 0.8783 into a calculator and finding the inverse sine, sin^-1(0.8783) = 57 degrees, then take the cos(57) degrees, which = 0.5446. Put a negative sign in front, to get -0.5446, since the angle is in quadrant II. Cosines are all negative in quadrant II.