The angles 180° − θ, 180° + θ, 360° − θ and −θ: reflections in the axes and in the origin
Take an acute angle θ and its point P(x, y) on the unit circle. Reflect P in the y-axis: you get (−x, y), the point of 180° − θ. Reflect in the origin: (−x, −y), the point of 180° + θ. Reflect in the x-axis: (x, −y), the point of 360° − θ and also of −θ. Reading sin = y, cos = x and tan = y/x from these points gives the table. The name of the function stays the same; only the sign may change, and the sign is the sign of that function in the quadrant. So to find sin 150°, write 150° = 180° − 30°, it is in the second quadrant where sin is positive, and sin 150° = sin 30° = ½.
| Angle | Quadrant | sin | cos | tan |
|---|---|---|---|---|
| 180° − θ | 2nd | sin θ | −cos θ | −tan θ |
| 180° + θ | 3rd | −sin θ | −cos θ | tan θ |
| 360° − θ | 4th | −sin θ | cos θ | −tan θ |
| −θ | 4th | −sin θ | cos θ | −tan θ |
Mirror points. On squared paper draw the unit circle (radius 5 cm). Mark the point for 30°, then fold the paper along the y-axis to find the point of 150°, then along the x-axis for 330°. Read the coordinates and compare with the table.