Express 6sin(2x)+5cos(x) in the form Rsin(x+a) (0degrees<x<90degrees)
Expand sin(2x) to 2sin(x)cos(x) so you have 6(2sin(x)cos(x))sec(x). cos(x) and sec(x) cancel out so the expression becomes 12sin(x) + 5cos(x). compare this to the expanded version of Rsin(x+a) which is Rsin(x)cos(a) + Rcos(x)sin(a) and we can see that Rcos(a) = 12 and Rsin(a) = 5. sin/cos = tan therefore we divide Rsin(a) = 5 by Rcos(a) = 12, giving us tan(a) = 5/12. we can then solve this equation on a calculator to give us the value of a=22.62 degrees.The next step is to use SoH & CaH to determine the value of R if you compare Rcos(a)=12 to CaH and Rsin(a)=5 to SoH. When this is visualized on a right angle triangle. You have an angle of 22.62 degrees, the line opposite to that angle will have a length of 5 and the line adjacent has a length of 13, with the hypotenuse having a length of R. Pythagoras' theorem can then be used as R2 = 132 + 52. This gives us the value of R = 13. so subbing our values back in. we get the answer of:6sin(2x)sec(x) + 5cos(x) = 13Sin(x + 22.62)
