I want to find all six trig function values, at -7 pi over 6. Now the first step in doing a problem like this is to draw a diagram, and you want to draw a diagram of your angle -7 pi over 6 to see what quadrant it's in, to see what it looks like, and you also want to be able to find the reference angle.
So reference angle is the angle between the terminal side, and the x axis and in this case it's pi over 6. Now the second step is to find the sine, cosine and tangent of your reference angle, pi over 6. Now you will recall that, the sine of pi over 6 is Â½, and the cosine of pi over 6 is root 3 over 2, and the tangent of pi over 6. If you don't remember, you can always just take the quotient Â½ over root 3 over 2, it's one over root 3 or root 3 over 3.
Now in order to get the sine, cosine and tangent at -7 pi over 6, I have to decide whether the sine, cosine and tangent, should be positive or negative in this quadrant. Now we're in the second quadrant, and this quadrant, all the x coordinate is going to be negative, and the y coordinates is going to be positive. So sine should be positive here. The cosine and tangent should be negative.
So since sine is positive, it's just going to equal Â½. Now cosine and tangent will be negative, so cosine of -7 pi over 6, negative root 3 over 2, and tangent negative root 3 over 3.
Now to get the reciprocal trig function values, all I have to do is take the reciprocals of these values. Remember the reciprocal of sine is cosecant. So to get the value of cosecant of -7 pi over 6, I take the reciprocal of Â½ which is 2.
To get secant, I take the reciprocal of cosine. Secant of -7 pi over 6 is 1 over negative root 3 over 2, and that's -2 over root 3 which is the same as -2 root 3 over 3. Finally for cotangent, I take the reciprocal of tangent, cotangent of -7 pi over 6 is the reciprocal of negative root 3 over 3 and that's negative root 3.
So just to recap; first draw a diagram. Find the reference angle. Find the sine, cosine and tangent of the reference angle, then use the quadrant to decide whether the sine, cosine and tangent are positive or negative. And finally, use the reciprocal identities to find the values of cosecant, secant and cotangent.