What are other examples

Question

 

5. While Link Concepts was playing with his calculator he found that sometimes the

answers produced from taking the sine and cosine of different angles were the same

answer. The information below shows some of the examples of when he found this to

occur.

sin 55° = 0.819152044 and cos 35° = 0.819152044

sin 30° = 0.5 and cos 60° = 0.5.

sin 15° = 0.258819045 and cos 75° = 0.258819045

Give Link some examples of other times when this occurs. Also, use your knowledge of

trigonometry to explain to Link why these calculations produce the same answer

Answer 1

Other examples:

sin(0)=cos(90°)=0; sin(90°)=cos(0); sin(45°)=cos(45°)=0.7071 (1/√2 or √2/2).

The reason comes from the word "trigonometry" which comes from "trigon" (meaning a 3-sided polygon, which is a triangle) and "metr-" (meaning measure). The "trigon" (triangle) is a right triangle so the other two angles in the triangle must add up to 90°.

sine=(opposite side length)/(hypotenuse length);

cosine=(adjacent side length)/(hypotenuse length).

What is the opposite side for sine is the adjacent side for cosine, and vice versa, hence why:

cos(x)=sin(90°-x) for angles between 0 and 90°. It's still true for angles bigger than 90° and negative angles, but it's easier to see why cosine and sine are related by looking at an "ordinary" right triangle where the relevant angles are each less than 90 degrees.