Derive! The the Sum and Difference Formulas/Identities for Sine and Cosine (MathAngel369)

The Derivation of the Sum and Difference Trig Identities/Formulas (MathAngel369)



In this discussion, we are going to derive the sum and difference identities, also known as the sum and

difference formulas, for sine and cosine.

Here is an outline of this video:
(Skip ahead, if you like)

🔹Review of the Fundamental Trig Identities
(Skip to 0:30)

🔹Deriving the Difference Formula for Cosine from the Unit Circle
(Skip to 0:52)

🔹Deriving the Sum Formula for Cosine
(Skip to 14:40)

🔹Deriving the Sum Formula for Sine
(Skip to 16:49)

🔹Deriving the Difference Formula for Sine
(Skip to 20:34)

It is beneficial to have a good algebra background before watching this video. However, I plan to show all

steps; so hopefully, you will find this video easy to follow regardless of your algebra background.

We will be using the fundamental Trigonometric Identities. If you need to review the fundamental trig

identities, I will display them in the video.
Okay, let us begin!


🔹Deriving the Difference Formula for Cosine


We begin with a unit circle.


Let the center of the circle be point O.



Since we are on the unit circle, the positive x-axis has terminal points (1,0) and we will call this point P.


Now draw an angle B anywhere you like. For this discussion, I will draw it somewhere in quadrant I.


The angle B has terminal points (cosB, sinB) and let’s call this point Q.


Next, draw an angle A wherever you like. For this discussion, I will draw it in quadrant II.


The angle A has terminal points (cosA, sinA) and let’s call this point R.



Now, if we begin at A and take away B, we have the angle (A-B).

So, I draw the angle (A-B) and it has terminal points (cos(A-B), sin(A-B)) and let’s call this point S.



Thus, from the positive x-axis to A-B, we have an angle A-B.

In other words,

the points S-O-P make the angle A-B.

Note that all points, from the center to the circle, has radius one since we are using the unit circle.

Now put on your thinking caps here!


If I begin at the positive x-axis, and take away B,

then from B to A we also have the angle (A-B).

In other words,

the points R-O-Q make the angle (A-B).

Let me restate this:

The points S-O-P make (A-B)

and

the points R-O-Q make (A-B).

This means that the distance between P and S is equal to the distance between Q and R.


So, we draw a line from P to S, and draw another line from Q to R.



Now we can use the distance formula that we learned from Precalculus Algebra.

Recall:

The distance formula says that, given a point U(x1, y1) and V(x2, y2,), the distance between U and V is

Sqrt [ ( x2 - x1 )^2 + ( y2 - y1 )^2 ].


So let us take the distance between the points

P(1,0) and S(cos(A-B), sin(A-B))


and set it equal to the distance between the points

Q(cosB, sinB) and R(cosA, sinA).


After some algebraic manipluation, we wil have the difference formula for cosine (see work in video).



🔹Deriving the Sum Formula for Cosine


Now we use the difference formula for cosine to find the sum formula for cosine (see work in video).



🔹Deriving the Sum Formula for Sine



We use the cofunction identities and the difference formula for cosine to derive the sum formula for sine (see work in video).



🔹Deriving the Difference Formula for Sine

We use the sum formula for sine to find the difference formula for sine (See work in video).



We have derived the sum and difference formulas for sine and cosine!

Thank you for watching!

I hope that I can assist you on your math journey!

Sincerely,

MathAngel369

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