As cos θ = x and sin θ = y in the unit circle. Therefore, depending upon the signs of x and y in different quadrants, the signs of trigonometric ratios change. The adjoining figure shows signs of trigonometric ratios in different quadrants.
Diagram image pending upload by Admin.
Trigonometric Ratios between 90° and 180°
We have discussed finding the sine and cosine for angles in the first quadrant, but what if our angle is in second quadrant (90°< θ < 180°)? In the diagram, R(1,0) and P(a,b) are points on the unit circle. Under a reflection in the y- axis, the image of P(a,b) is P′(- a,b) and the image of R(1,0) is R′(- 1,0). Since angle measure is preserved under a line reflection, ∴ ∠ROP = ∠R′OP′ Also ∠ROP′ and ∠R′OP′ are supplementary angles. ∴ ∠ROP′ + ∠R′OP′ = 180° (i) But, ∠R′OP′ = ∠ROP Therefore, (i) becomes: ∠ROP′ + ∠ROP = 180° ∠ROP = 180° - ∠ROP′ = 180° - θ