If a_b tan x=c/cosx, then findthat a sin x +b sin x =?
Answers
{\displaystyle f(x)=a\sin(x)+b\cos(x)}
We shall show that this is a sinusoidal wave, and find its amplitude and phase.
To make things a little simpler, we shall assume that a and b are both positive numbers. This isn't necessary, and after studying this section you may like to think what would happen if either of an or b is zero or negative.
We'll first use a geometric argument that actually shows a more general result, that:
{\displaystyle g(\theta )=a_{1}\sin(\theta +\lambda _{1})+a_{2}\sin(\theta +\lambda _{2})} {\displaystyle g(\theta )=a_{1}\sin(\theta +\lambda _{1})+a_{2}\sin(\theta +\lambda _{2})}
is a sinusoidal wave. Since we can set {\displaystyle \lambda _{1}=0^{\circ }\ ,\ \lambda _{2}=90^{\circ }} {\displaystyle \lambda _{1}=0^{\circ }\ ,\ \lambda _{2}=90^{\circ }} the result we are trying for with {\displaystyle f} f follows as a special case.
We use the 'unit circle' definition of sine. {\displaystyle a_{1}\sin(\theta +\lambda _{1})} {\displaystyle a_{1}\sin(\theta +\lambda _{1})} is the y coordinate of a line of length {\displaystyle a_{1}} a_{1} at angle {\displaystyle \theta +\lambda _{1}} {\displaystyle \theta +\lambda _{1}} to the x axis, from O the origin, to a point A.
If we now draw a line {\displaystyle {\overline {AB}}} \overline {AB} of length {\displaystyle a_{2}} a_{2} at angle {\displaystyle \theta +\lambda _{2}} {\displaystyle \theta +\lambda _{2}} (where that angle is measure relative to a line parallel to the x axis), its y coordinate is the sum of the two sines.
However, there is another way to look at the y coordinate of point {\displaystyle B} B . The line {\displaystyle {\overline {OB}}} {\displaystyle {\overline {OB}}} does not change in length as we change {\displaystyle \theta } \theta , because the lengths of {\displaystyle {\overline {OA}}} {\displaystyle {\overline {OA}}} and {\displaystyle {\overline {AB}}} \overline {AB} and the angle between them do not change. All that happens is that the triangle {\displaystyle \Delta OBC} {\displaystyle \Delta OBC} rotates about O. In particular {\displaystyle {\overline {OB}}} {\displaystyle {\overline {OB}}} rotates about O.
This then brings us back to a 'unit circle' like definition of a sinusoidal function. The amplitude is the length of {\displaystyle {\overline {OB}}} {\displaystyle {\overline {OB}}} and the phase is {\displaystyle \lambda _{1}+\angle BOA} {\displaystyle \lambda _{1}+\angle BOA} .