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In square ABCD, P is the midpoint of \(\overline{BC}\) , and Q is the midpoint of \(\overline{CD}\). Find \(\sin \angle PAQ\)

 

Thank you!

 Oct 12, 2020
 #1
avatar+240 
+2

The sine of angle PAQ is 0.6

 Oct 12, 2020
 #4
avatar+240 
+2

By using the real numbers, we can simplify this simple question (and answer) even more.smiley

 

Let a side of the square be 2 units

 

(I'll use hecticar's diagram)

 

1/    ∠a ≅ ∠c = arctan(DQ / AD) ≈ 26.565º

 

2/    ∠b = 90º - (∠a + ∠c) ≈ 36.87º

 

3/     sin(∠b) = 0.6    or   3/5 

jugoslav  Oct 12, 2020
edited by jugoslav  Oct 13, 2020
 #7
avatar+111124 
-1

Why have you edited this?

Your answer was trivial and useless as most of your answers are.

Why don't you just be honest and be clear about your edits.

OR better still, make a new post if you have something quite different to say.

Melody  Oct 14, 2020
edited by Melody  Oct 14, 2020
 #2
avatar+9041 
+3

 

Let   m∠DAQ  =  a

Let   m∠PAQ  =  b

Let   m∠PAB  =  c

 

Notice that    △DAQ  ≅  △BAP,    and so    m∠DAQ  =  m∠PAB    which means    a  =  c

 

Then since the measure of an interior angle of a square is 90°...

 

a + b + c  =  90°

                               Since  a = c  we can substitute  a  in for  c

a + b + a  =  90°

                               Combine like terms

b + 2a  =  90°

                               Subtract  2a  from both sides to solve for  b

b  =  90° - 2a

 

And so....

 

sin(b)   =   sin(90° - 2a)

                                             Now we can use the rule that says:  sin(90° - θ)  =  cos(θ)

sin(b)    =   cos(2a)

                                             Now we can use the double-angle formula for cos

sin(b)   =   1 - 2sin2(a)

                                             And  sin(a)  =  opposite / hypotenuse  =  1 / sqrt(5)  **

sin(b)   =   1 - 2(1/5)

 

sin(b)   =  3/5

 

 

**If you would like more explanation on this part please feel free to ask!

 Oct 12, 2020
 #3
avatar+19 
0

Hello there hectictar, thank you so much for the reply, I've been stuck on this 1 for a while and I cant show enough support, I just have 1 question. how do we know that sin B is equal to sin(90 degrees+2a)? Thank you so much

Age26  Oct 12, 2020
 #5
avatar+9041 
+2

Do you see how we know that

 

b  =  90° - 2a   ?

 

(If you're confused on that part, let us know!!)

 

Then once we know that is true, there are a couple ways to think of the next step...

 

Given that:

 

b  =  90° - 2a

 

We can take the sin of both sides of that equation to get:

 

sin(b)  =  sin(90° - 2a)

 

Also......a kind of another way to think of this....

 

Looking at the expression:

 

sin(b)

 

Since  b = 90° - 2a,  we can substitute  90° - 2a  in for  b  to get the following equivalent expression:

 

sin(90° - 2a)

 

And so....

 

sin(b)  =  sin(90° - 2a)

hectictar  Oct 13, 2020

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