Question

Use one of the identities
cos(t + 2πk) = cos t or sin(t + 2πk) = sin t
to evaluate each expression. (Enter your answers in exact form.)
(a)
sin(19π/4)



(b)
sin(−19π/4)



(c)
cos(11π)



(d)
cos(53π/4)



(e)
tan(−3π/4)



(f)
cos(π/4)



(g)
sec(π/6+ 2π)




(h)
csc(2π − 2π/3)

Answer 1

here is how I do these ....
(perhaps you might find it easier in degrees, 19π/4 radians = 855°

sin(19π/4) = sin 855°
take away 2 rotations, (720° or 4π)
sin 19π/4 = sin 3π/4 = sin 135°
3π/4 or 135° is in quadrant II and π/4 or 45° from the x-axis
We also know that in II, the sine is positive
so sin 3π/4 = sin π/4 = sin 45° = 1/√2 or √2/2

.....

d) cos(53π/4) or cos 2385°
take away 12π or 2160°
cos 53π/4 = cos 5π/4 or cos 225°
This angle is in quad III making it negative and 45 from the x-axis

cos 53π/4 = - cos π/4 = -1/√2 or -√2/2

do the others the same way.

Answer 2

cos11pi/6