Question

@tranquility

Answer 1

sin(A+b) = sinAcosB + cosAsinB

Answer 2

how do i prove that lmao

Answer 3

It's really weird that they're asking you to prove that since it's one of our core identities. And it's not proved the same way we were were proving your other identities.

This webpage has the proof for that identity. If you don't understand something, we can go over it.
https://themathpage.com/aTrig/sum-proof.htm

Answer 4

I may or may not understand anything from that?

Answer 5

Let's do it one step at a time. Since you said you had a unit circle, I'll just be following this
https://services.math.duke.edu/~leili/teaching/uwm/math222s11/problems/quizzes/trig.pdf



Following along still?

Answer 6

We have a right angle triangle and the angle opposite AE is (a+b)

Answer 7

oh, yea

Answer 8

Looking at our current diagram, what is \(\sin(a+b) = \)

Answer 9

AE

Answer 10

Since it's a unit circle, that means from the origin to A, the distance is 1.

Answer 11

yea, ik that in unit circle radius=1

Answer 12

Do you see that AE = AD + DE

Answer 13

yea, I see it
altho, where'd the box come from-

Answer 14

We drew a line from angle a to the end of the unit circle and then drew two lines from it- one to mark a new point C and another to mark D

Answer 15

It's just to help us prove this identity

Answer 16

oh, kayyy

Answer 17

We're going to draw another line and make a new triangle. We want to find the value of AD



angle BAD is equal to angle a because they're alternate angles.

Answer 18

how are they alternate angles...?

Answer 19

Because the lines DB and OC are parallel and we have a straight line OB going through them

Answer 20

how does that relate to angle BAD

Answer 21

sorryyyyyyyyyyyyyyyyyyyy
ig now i can never complain about other ppl being slow anymore-

Answer 22

Just believe it

okay so
angle ABO is 90 degrees

We know that angle OBD is angle A
(you should be able to see clearly that it's equal to a since they're vertical angles)

Triangle ADB has three angles one of which is a 90 degree angle. That means the remaining two angles are complementary.

Answer 23

okay, yea, I get that

Answer 24

Well, I guess I should have mentioned that AB is perpendicular to OB
And that BC is perpendicular to OC

Answer 25

That's how the angle in the corner is equal to a

Answer 26

oh, kay

Answer 27

Are you really sure that your teacher asked you to prove this? This is some complicated stuff even for a 10th grade geometry course

Answer 28

I'm in 11th grade math, trig

Answer 29

okay so now we go back to before we tried to prove that it was indeed angle a

Answer 30

wat

Answer 31

does this mean she's going to explain it tomorrow

Answer 32

I mean knowing the proof of this identity is pretty useless as long as you memorize the formula

Answer 33

Trying to derive this identity takes way too much time anyway

Answer 34

We said that AE = AD + DE

Now let's determine what AD is

AD=AB*cos(BAD)

We said that angle BAD is actually just angle a so we get

AD =AB*cos(a)

Since we're trying to prove the identity, how would you write AB in terms of sin and angle b

Answer 35

sin(b) = (AB*cos(a))/AB
AB = (AB*cos(a))/sin(b)

Answer 36

I have no idea what you did there

Look at the triangle


What is sin(b)

Then solve for AB

Answer 37

sin(b) = AB

Answer 38

Exactly. Let's put it all together now.

We said

sin(a+b) = AE = AD + DE

and that AD = AB * cos(a)

and that AB = sin(b)

Now we get

AD = sin(b) * cos(a)

and we can plug that into get

sin(a+b) = AE = sin(b)cos(a) + DE

Answer 39

Now we need to solve for DE.

Do you see how DE = BC in our image?

Answer 40

yea

Answer 41

So then I have to find sin(a)?

Answer 42

If DE = BC

We can say that

BC = OB * sin(a)

Do you agree?

Answer 43

yep

Answer 44

Now we have

sin(a+b) = AE = sin(b)cos(a) + (OB * sin(a))

But that isn't good enough. Let's re-write OB in terms of cos and angle b so that it matches our identity

Answer 45

cos(b) = OC/BD
OC = cos(b) * BD

Answer 46

We said sin(b) = AB
What is cos(b) = ??

Answer 47

cos(b) = OB

Answer 48

right, i keep on forgetting radius is 1

Answer 49

\(\color{#0cbb34}{\text{Originally Posted by}}\) @snowflake0531
cos(b) = OC/BD
OC = cos(b) * BD
\(\color{#0cbb34}{\text{End of Quote}}\)

That's actually an O for origin

And this won't get us very far so we use the other triangle since OA is 1

Answer 50

ohh, kay

Answer 51

Because both OB and OC does not equal 1 so the other triangle helps us get what we're looking for

Answer 52

Since OB = cos(b)

Now we get

sin(a+b) = AE = sin(b)cos(a) + cos(b)sin(a)

Answer 53

And that is how you derive the identity

Answer 54

ohhhhhhh kay
xdddddddddd
thank you for being so patient lmao
<3

Answer 55

No problem!