Confused about proof of "sin(θ + Φ) = cosθsinΦ + sinθcosΦ"

[JS-Student]

Hi,
This is also a sort of geometry question.
My textbook gives a proof of the relation: sin(θ + Φ) = cosθsinΦ + sinθcosΦ.
It uses a diagram to do so:

http://imgur.com/gLnE2Fn

sin (θ + Φ) = PQ/(OP)
= (PT + RS)/(OP)
= PT/(OP) + RS/(OP)
= PT/(PR) * PR/(OP) + RS/(OR) * OR/(OP)
= cosθsinΦ + sinθcosΦ

My confusion with this is
How do they know that angle TPR also measures θ?
How do they know that triangle TPR is similar to triangle ROQ?

Thanks

The textbook is: Calculus with Analytic Geometry, 2e by George F. Simmons

 

[thegirl]

Hey,

I think I kind of figured it out, angle ros = theta, angle ors = 90 -θ

the angle between the TR and the brown line is theta, the angle between PR and the brown line is 90degrees right so angle PRT is 90 - theta and so TPR is theta I've attached a diagram because I feel like these words aren't making sense. Is the diagram clear?

They are similar triangles because they have the same angles.

 

[SammyS]

Hi,
This is also a sort of geometry question.
My textbook gives a proof of the relation: sin(θ + Φ) = cosθsinΦ + sinθcosΦ.
It uses a diagram to do so:
View attachment 90905
http://imgur.com/gLnE2Fn

sin (θ + Φ) = PQ/(OP)
= (PT + RS)/(OP)
= PT/(OP) + RS/(OP)
= PT/(PR) * PR/(OP) + RS/(OR) * OR/(OP)
= cosθsinΦ + sinθcosΦ

My confusion with this is
How do they know that angle TPR also measures θ?
How do they know that triangle TPR is similar to triangle ROQ?

Thanks

The textbook is: Calculus with Analytic Geometry, 2e by George F. Simmons

They use geometry.

I assume they intend for ∠ORP to be a right angle.

∠RTO measures θ. ∠TRP measures 90° - θ . etc.

(I assume you meant ΔROS, not ΔROQ .)
As for ΔTPR and ΔROS, they're both right triangles each having an acute angle with the same measure, namely θ.

 

[JS-Student]

Hey,

I think I kind of figured it out, angle ros = theta, angle ors = 90 -θ

the angle between the TR and the brown line is theta, the angle between PR and the brown line is 90degrees right so angle PRT is 90 - theta and so TPR is theta I've attached a diagram because I feel like these words aren't making sense. Is the diagram clear?

They are similar triangles because they have the same angles.

Oh, wow thanks. It makes sense now. Thanks especially for taking the time to upload a picture.