If sin(A)=1213sin⁡(A)=1213 and is in Q2 and sin(B)=−35sin⁡(B)=−35 in Q4, what is sin(A2)sin⁡(A2) and cos(A−B)cos(A−B)?

Answer 1

If you form a right triangle in the 2nd quadrant with angle a off the horizontal, label the base -5, the height 12, and the hypotenuse 13.

sin a=12/13

cos a =-5/13

Make another right triangle this one in the 4th quadrant. The angle off the horizontal is b. You should label the hypotenuse 5, the vertical side -3, and the horizontal side 4. From this

sin b = -3/5

cos b = 4/5

The question asks for what is sin(a/2)?

sin(a/2)=+ or - sqrt[(1-cosa)/2]

Plug in the value of cosa

+/- sqrt[(1- -5/13)/2]

+/- sqrt[(1+5/13)/2]

+/-sqrt[(13+5)/13*2]

+/-sqrt[18/26]

+/-sqrt(9/13)

+/-3/sqrt 13

Multiply top and bottom by sqrt of 13

+/[3sqrt13]/13

Since sin is positive in the 2nd quadrant, the answer should be only the positive value +[3sqrt13]/13

cos(a - b)= Cos(a)cos(b) + sin(a)sin(b)

Plug In values

(-5/13)(4/5)+(12/13)(-3/5)

(-20/65)+(-36/65)

-56/65