sin t = 4/5 find sin(π - t) and find sin(t + π)?

Question

My math teacher told me to find these but I cannot figure it out. Can someone help me? I have tried a whole bunch of different ways, including changing to decimals, and radians etc., but I cannot find how to get the answer.

I know the answers are 4/5 and -4/5, but I need to know how to get there.

Answer 1

sin(π - t) = sin t = 4/5
sin(t + π) = -sin t = -4/5
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Reason: In a unit circle, sin(Θ) is the y coordinate. At the positions of t and π-t, you have the same y coordinate. At the positions of t and π+t, you have the opposite y coordinates. Or you can use double angle formula to verify it.
sin(A+B) = sinAcosB + cosAsinB
sin(π - t) = sinπcost - cosπsint = sint
sin(π + t) = sinπcost + cosπsint = -sint

Answer 2

It might help if you have the graph of sin sitting in front of you.

First, note that it is antisymmetric about y, e.g. sin is odd, and sin(-x) = -sin(x)

Then notice that if you shifted the the graph right by pi, eg sin(x) ---> sin(x-pi) (or left by pi, for that matter) that the new plot is just upside down, or -sin(x).

Using this knowledge, sin(pi - t) = -sin(t-pi) = sin(t)
and sin(t + pi) = -sin(t)