[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.
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.
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!