Transforming sinx to sinx + cosx

[maxim07]

Homework Statement

I need to transform sinx to sinx + cosx

Relevant Equations

Trig identities I’ve used are sin^2x + cos^x = 1 and sin2x = 2sinxcosx

y = sinx

stretch by scale factor 1/2 in x direction

y = sin2x

translation in y direction by 1

y = 2sinxcosx + 1

= sin^2x + 2sinxcosx + cos^2x

= (sinx + cosx)^2

I don’t know whether you can get rid of a square with a transformation

 

[Mark44]

Homework Statement:: I need to transform sinx to cosx
Relevant Equations:: Trig identities I’ve used are sin^2x + cos^x = 1 and sin2x = 2sinxcosx

y = sinx

stretch by scale factor 1/2 in x direction

y = sin2x

translation in y direction by 1

y = 2sinxcosx + 1

= sin^2x + 2sinxcosx + cos^2x

= (sinx + cosx)^2

I don’t know whether you can get rid of a square with a transformation

What exactly are you trying to do? The thread title says transforming sin(x) to sin(x) + cos(x), but in the statement above, it says "I need to transform sinx to cosx".

If it's the latter, ##\sin(x) = \cos(\pi/2 - x)##.

 

[maxim07]

My mistake, a typo in the homework statement, the thread title is correct in saying sinx to sin2x + cosx. I have amended it now.

 

[maxim07]

Anyone got any idea how to map sinx onto sinx + cosx via a transformation?

 

[phyzguy]

I'm not sure exactly what you're trying to do, but try using sin(x+y) = sin(x)cos(y) + cos(x)sin(y) and choose y appropriately.

 

[maxim07]

The question requires me to perform a geometrical transformation such as a translation, stretch or reflection, that turns sinx into sinx + cosx. The question asks for a sequence of transformations, so maybe it requires more than one transformation.

Using sin(x+y) my first guess would be to make sin(y) and cos(y) equal1, so that I am left with sinx + cosx
but there isn’t a value of y where both equal 1

The only part of the graphs where sin(y) = cos(y) is when y = π/4 (as far as I can tell)

this leaves me with sin(x + π/4) = sin(x)cos(π/4) + cos(x)sin(π/4) (translation in x direction by -π/4)

= 0.707...sin(x) + 0.707...cos(x)

if I use a stretch in y direction by scale factor 1/0.707... I’ll get sin(x) + cos(x)

maybe this is what is expected

 

[haruspex]

maybe this is what is expected

Looks right.