Calculate sin13pi/3, cos33pi/4 and sin101pi/2
Solution
We'll calculate sin13pi/3.
We notice that we can write:
13pi/3 = pi/3 + 12pi/3
pi/3 + 12pi/3 = pi/3 + 4pi
But 4pi = 2*2pi. We can substitute 2pi by 0, because 2pi and 0 are overlapping.
So,
sin 13pi/3 = sin (2*2pi + pi/3)
sin (2*2pi + pi/3) = sin pi/3
sin13pi/3 = sin pi/3 = sqrt3/2
We'll calculate cos33pi/4.
We notice that we can write:
33pi/4 = pi/4 + 32pi/4
pi/4 + 32pi/4 = pi/4 + 8pi
But 8pi = 4*2pi
We'll substitute 2pi by 0.
4*2pi = 4*0 = 0
cos33pi/4 = cos (pi/4 + 4*2pi)
cos (pi/4 + 4*2pi) = cos pi/4
cos33pi/4 = cos pi/4 = sqrt2/2
We'll calculate sin 101pi/2.
We notice that we can write:
101pi/2 = pi/2 + 100pi/2
pi/2 + 100pi/2 = pi/2 + 50 pi
But 50pi = 25*2pi
We'll substitute 2pi by 0.
sin 101pi/2 = sin (pi/2 + 25*2pi)
sin (pi/2 + 25*2pi) = sin pi/2
sin 101pi/2 = sin pi/2 = 1