Help with trig identities? ?

Question

I've forgotten a lot of the trig identities, so someone please tell me how to do problems like these (without using calculator, may use unit circle):

tan^-1(-1) ; sin^-1 (-1); sin^-1 (sqrt(3)/2)

Answer 1

Apart from understanding the unit circle (which I presume you remember), the key to these is to remember the two paradigm triangles from geometry.  The 45-45-90 and 30-60-90 triangles are the primary ones to give you exact values for trig functions.

For instance:  tangent is opposite over adjacent.  If the tangent is 1, then the opposite and the adjacent must be of equal length.  That means 45-45-90, or an acute angle of pi/4.  For arctan(-1), the angle must be in the second or fourth quadrant (to make the tangent negative) and the reference angle must be pi/4.  So, arctan(-1) = -pi/4.

arcsin(-1) puts you at -pi/2.  Just sketch the unit circle and look where the sine is (a) negative and (b) 1.

arcsin(sqrt(3)/2)) involves the 30-60-90 triangle.  Sine is opposite over hypotenuse, and that sqrt(3) is a dead giveaway.  If you sketch the triangle, we're obviously talking about the 60 degree angle, or pi/3, and the fact that the sine is positive puts us in the first quadrant.  So, arcsin(sqrt(3)/2) = pi/3.

Hope that helped.