1. First, we should determine the quadrant in which 60∘ lies.
It is obviously the first quadrant. Then, we should recall whether sine in the first quadrant is positive or negative. "All Silver Tea Cups" tells us that all the trigonometric ratios are positive in the first quadrant; therefore, sin60∘ must be positive.
Now, we should write 60∘ in such a way that it is ±30∘ with any of the two axes (the horizontal XOX′ and the vertical YOY′) . So we can write sin60∘=sin(90∘−30∘). Now, we can recall from the trigonometric ratios in the previous page that sin(90∘−θ)=cosθ
⇒sin60∘=sin(90∘−30∘)=cos30∘=√3/2.
2. Similarly, we can find out the value of sin120∘. This angle lies in the second quadrant. In the second quadrant, sine is positive. Therefore, sin120∘= some positive value.
sin120∘=sin(90∘+30∘)=cos30∘=√3/2
3. sin240∘ lies in the third quadrant. So it should be negative.
sin240∘=sin(270∘−30∘)=−cos30∘=−√3/2
4. sin300∘ lies in the fourth quadrant. So it should be negative.
sin300∘=sin(270∘−30∘)=−cos30∘=−√3/2
5. sin(−30∘) lies in the fourth quadrant. So it should be negative.
sin(−30∘)=−sin30∘=−1/2.