The Matrix Reloaded: Deriving sin(A+B), cos(A+B) by Rotation Matrices

In our previous post we asked students to verify the sin(A+B) identity for an angle of 75°. Although one might try to generalize the result, there are many other derivations for the sum and difference identities that teachers have seen or used. Those who teach this topic whether it be in an Advanced Algebra/Trig, Advanced Math, Precalculus, or some similarly named course have the choice of deriving the formula, outlining a proof and having students attempt to provide the details or simply motivating the formula. My guess is that, unless you have a highly motivated and strong group of students, the full derivation is not done in class. By the way, an excellent applet for helping students visualize these formulas is located here. One could use this for classroom demonstration or in tutorial mode.

The forms of these trig identities often engage students and some may wonder if there's a pattern one can recognize that occurs elsewhere in mathematics. Using nonsense names for functions, these rules seem to have the form: the squiggle of the first times the squeegee of the second ± the squiggle of the second times the squeegee of the first. Students will hopefully recognize this pattern again when they see the product rule in calculus. Is there an overarching concept that encompasses forms like this? Well, I'd like to suggest one in this post. You can decide for yourself if it's worth developing the required machinery for secondary students (or it can be an enrichment topic or project).

The trig identities mentioned above have the form of a sum (or difference) of products. Where might one encounter this in mathematics? For myself, it's when I studied the products of matrices or the dot product of 2 vectors. v₁⋅v₂ = a₁b₁+a₂b₂. Matrix products and vectors are required in some states' curricula (CA for example), so there is a basis for this approach (pun intended!!).

To simplify , we will focus on position vectors with a length of 1 so that their terminal points are all on the unit circle. Students learn early on in trig that each point on the unit circle can be represented using the parameter t or θ as in (cosθ, sinθ) and that θ represents an angle or rotation, counterclockwise for θ>0, etc. Another way of viewing this is that the unit vectors (1,0) and (0,1), (which are labeled i and j by most physics teachers), are transformed by this rotation as follows:
(1,0) ---> (cos θ, sin θ) (*)
(0,1) ---> (-sin θ, cos θ).
These results are straightforward and students should be able to demonstrate them graphically for values of θ in Quadrant I. For those who recall the terminology from linear algebra, (1,0) and (0,1) are referred to as an orthornormal basis.

[We will use the notation R_θ to denote the rotation transformation by an angle θ about the origin]:

STUDENT (or reader) ACTIVITY
1. Have students verify or find the following (either graphically or using (*) above) :
R_90° (1,0) = (0,1)
R_90° (0,1) = (-1,0)
R_60° (1,0) = (1/2,√3/2)
R_60° (0,1) = ______
R_45° (1,0) = ______
R_45° (0,1) = ______

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We will now show how the following 2x2 rotation matrix can accomplish this transformation:
(**)

Note that the first column (vector) is the rotation image of the vector (1,0) and similarly for the 2nd column. This idea of using (orthonormal) basis vectors to construct a matrix representing a transformation is crucial in linear algebra. This requires more development and that is not the purpose of this post. We are asking students to accept this for now, although we will have them verify that this matrix approach produces correct results in specific instances. We will now represent the rotation image of any point on the unit circle by multiplying this matrix by the column vector which represents that point. In rectangular (x,y) notation:

(***)

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STUDENT ACTIVITY (cont.)

2. Write the matrix form (**) of R_θ for θ =
(a) 90°
(b) 60°
(c) 45°
(d) -90°

3. Re-do problem #1 above using the matrix multiplication formula (***). Make sure your results match!
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So how does all of this relate to the sum/difference trig identities? We're almost there! Instead of using (x,y) to represent an arbitrary point on the unit circle, we will use the trigonometric form (cosα,sinα) in column vector form. For now we will assume α is between 0 and π/2:

STUDENT ACTIVITY (cont.)

4. Perform the following matrix multiplication:


Does the resulting column vector look familiar? Answer the next two questions and maybe you'll figure out why!

5. The result of #4 is equivalent to a single rotation of what angle?

6. What can we conclude?
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There's much more to say here but this is enough for now. As usual, I take full responsibility for errors or ambiguities. I await your thoughts. If this is a derivation you've seen before, let me know. One could also derive this using complex numbers or polar coordinates or ...

Another View of sin(A-B) for a Special Case: An Investigation

[As always, don't forget to give proper attribution when using this in the classroom or elsewhere as indicated in the sidebar]

In a standard trig unit, students learn those wonderful formulas for the sin and cos of the sum and difference of angles. Many creative methods have been developed to derive these formulas and, depending on the ability of the group and teacher preference, these are demonstrated or not. Students are typically shown various mnemonics for recalling them on the big test, but, in this investigation, students will derive sin 15° using only 30-60-90 triangle ratios and the Pythagorean Theorem. We will then compare the result to that obtained by the traditional formulas for sin(45°-30°) or sin(60°-45°) and show equivalence by algebraic methods using radicals. This is not an attempt to develop a general approach to deriving sum/difference formulas, although readers are invited to try a generalization. You may recall other posts on this blog of a similar nature.

THE PROBLEM/INVESTIGATION
Refer to the triangle above. If the print is too small, click on the image to magnify.
∠A = 75° and ∠B = 15°

(a) In the triangle above, locate point D on side BC such that ∠CAD = 60° . Express the lengths of the sides of triangle CAD in terms of a.
[Note: We could avoid the variable a altogether and assign a value of 1 since this is a ratio problem.]
(b) Show that CB = a √3 +2a.
(c) Use the Pythagorean Theorem to show that AB = a √(8+4 √ 3)
(d) Verify the identity (√ 6 + √ 2)² = 8+4 √ 3. Use this to rewrite AB.
(e) Use above results to obtain an expression for sin 15°.
(f) Use the standard trig formula for sin(45°-30°) to obtain an expression for sin 15°.
(g) Show your results in (e) and (f) are equivalent.

An Instructional Aside
When introducing the formula for sin(A+B), for example, teachers sometimes motivate it effectively using numerical values or considering the special case A=B. Here's an alternative:
Consider the special case A+B = 90°
Ask students to verify the formula for sin(A+B) in this special case. Simple, but at least it's something slightly different to pique their curiosity.

Standard Disclaimer:
This investigation is not copied from some other source. As it is original and has not been edited by others, there's always the possibility of error. Please feel free to suggest corrections/edits/extensions...
You know I welcome your comments!