Sin and Cos functions

Question

Without a calculator, find the exact value of:

 

a) sin(120)

 

b) cos(-3Π/4)

Answer 1

a)

 

Find the smallest acute angle for 120.  Since this angle is in the second quadrant,

 

180 - 120 = 60

 

sin(60) = √(3) / 2

 

Since sine is positive in the second quadrant,

 

sin(120) = √(3)/2

 

 

 

b)

 

-3π/4 is the angle rotated clockwise from the positive x-direction.  This means to get the positive angle, we subtract it from 2π.

 

2π - (3π/4) = (8π/4) - (3π/4) = 5π/4

 

 

This positive angle is in the 3rd quadrant with a reference angle of π/4.

 

cos(π/4) = √(2)/2

 

Since cosine is negative in this quadrant,

 

cos(-3π/4) = -√(2)/2

Answer 2

a)

reference angle is 60° (180° - 120°)

sin 120° = sin 60°

b) 

reference angle for -3π/4 is π/4 (π - 3π/4)

cos -3π/4 = -cos π/4