Expert: Scotto - 1/13/2009

Question
Write the expression as a single real number. Do not use decimal approximations.

A) SIN(Pi/6)Cos(PI/2) - Cos(pi/6)SIN(pi/2)

B) SIN(-13PI/6)COS(5pi/4)+ COS(-13pi/6)SIN(5pi/4)

Answer
You need to know the sin() and cos() of 30° and 60°.
30° is π/6 radians and 60° is π/3 radians.

If you think about a triangle with all sides and all angles equal, it is easy to deduce all the rest if one side is given to have length one.  Since all of the sides are equal, they all have length one.  Since all of the angles are equal and they add up to 180°, each of them is 180°/3 = 60°.

Put one of the points in the triangle at the top.  Now slice the triangle in half from the top down to where it is perpendicular with the base.  Since the angle has been sliced in half at the top, we know the angle on either side of the vertical line is 60°/2 = 30°.  We also know that the line hits the bottom at a right angle, if we look at the left side, we have a right triangle with angles (besides the right angle) of 30° and 60°.  The bottom side of this triangle is half of the length of the bottom.  The length of any side was one, so the length of the bottom of this triangle is 1/2.  From the Pythagorean theorem, we know that a² + b² = c² where a and b are the legs and c is the hypotenuse.  Since a is 1/2 and c is 1, we can say the 0.5² + b² = 1².  This tells us that b² = 1 - 0.25 = 3/4, so b = √3/2.  That takes care of 30° = π/6 radians and 60° = π/3 radians.

As far as the π/2, that's 45°.  Given one angle is 45° in a right triangle and that the sum of the two angle must be 90°, the other angle is 90° - 45° = 45°.  This can be seen to be half of a square (a square chopped from one corner to the opposite corner), with both sides being 1.  The hypotenuse has length c, and by the Pythagorean  theorem, we know that c² = 1² + 1², so c = √2.

Using these sides, you can find the sin() and cos() of the angles given.   Note that if the angle is between π/2 and 3π/2, the cos() is negative.  If the angle is between π and 2π degrees, the sin() is negative.  Try using these triangles to do the problem.