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March 26, 2017

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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)

  • Precalculus - ,

    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.

  • Precalculus - ,

    cos11pi/6

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