How Does cos(θ)sin(θ+φ)-sinθcos(θ+φ) Simplify to sinφ?

In summary, by using the trig identity sin(u+v) = sin(u)cos(v)+sin(v)cos(u) and the fact that sin(-u) = -sin(u), the expression cos(θ)sin(θ+φ)-sinθcos(θ+φ) simplifies to sinφ.
  • #1
leaf
4
0
Can someone please show me, step-by-step, how

cos(θ)sin(θ+φ)-sinθcos(θ+φ)

simplifies to sinφ?

I know I have to use trig identities, and I got to cos2θsinφ-sin2θcosφ but I'm not sure where to go from there.

Thanks
 
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  • #3
Yeah, if I substitute for both I get:

[itex]sin(\phi)-sin^2(\theta)sin(\phi)-cos(\phi)+cos^2(\theta)cos(\phi)[/itex]

But I should probably only substitute for sin or cos so that gives me:

[itex]sin(\phi)-sin^2(\theta)sin(\phi)-sin^2(\theta)cos(\phi)[/itex]

or

[itex]cos^2(\theta)sin(\phi)-cos(\phi)+cos^2(\theta)cos(\phi)[/itex]

but I'm still not sure where to go from here.
 
  • #4
sin(u+v) = sin(u)cos(v)+sin(v)cos(u). Also sin(-u) = -sin(u).
 
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  • #5
leaf said:
Can someone please show me, step-by-step, how

cos(θ)sin(θ+φ)-sinθcos(θ+φ)

simplifies to sinφ?

I know I have to use trig identities, and I got to cos2θsinφ-sin2θcosφ but I'm not sure where to go from there.

Thanks
Maybe someone else already told you but you can use the next identity

sin(u+-v) = sin(u)cos(v)+-cos(u)sin(v)

if oyu compare it with you homework then the answers is:

sin(θ+φ-θ)=sinφ
 
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Related to How Does cos(θ)sin(θ+φ)-sinθcos(θ+φ) Simplify to sinφ?

What does it mean to simplify an expression?

Simplifying an expression means to reduce it into a simpler form by combining like terms, using the order of operations, and eliminating unnecessary parentheses or terms.

Why is it important to simplify expressions?

Simplifying expressions allows us to solve equations and problems more easily by reducing the number of steps and making the expressions more manageable.

How do I simplify an expression?

To simplify an expression, follow the order of operations (PEMDAS), combine like terms, and eliminate unnecessary parentheses or terms.

What are some common mistakes when simplifying expressions?

Common mistakes when simplifying expressions include forgetting to follow the order of operations, incorrectly combining like terms, and not distributing negative signs correctly.

Can expressions with variables be simplified?

Yes, expressions with variables can be simplified by following the same steps as with numerical expressions. The variables are treated as placeholders and can be combined like terms or simplified using the order of operations.

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