7 Show that cos G sin 1 a Employing Pythagoras theorem Hint

7. Show that: cos G sin 1 a. Employing Pythagoras theorem. (Hint: refer to class notes b. Employing Euler\'s formula. Hint: use the results of Problem 5)

Solution

The traditional way to prove this (by geometrical trigonometry) is to draw a right-angle triangle ABC with the 90 degree angle at C.

The length of the sides are identified by lower case letters a (the side opposite angle A), b and c (the hypotenuse, since it is opposite the 90-degree angle at C).

Let\'s pick an angle (say, A)

sin(A) = a/c
cos(A) = b/c

sin²(A) + cos²(B) = (a/c)² + (b/c)² = a²/c² + b2/c²

same denominator, so we can factor it out:

sin²(A) + cos²(B) = (1/c²) (a² + b²)

By Pythagorean theorem, we have a² + b²= c²

sin²(A) + cos²(B) = (1/c²) (c²) = c²/c² = 1