The addition formulas in trigonometry present a recurring problem: we learn them, we forget them, we relearn them. Cos(a – b) and sin(a) appear in almost every senior year exercise and in the non-calculator sections of the baccalaureate. Understanding their logic rather than just memorizing them changes the game.
What the unit circle shows before any formula
Before manipulating cos(a – b), it is essential to visualize what happens on the unit circle. Place two points M and N on this circle, corresponding to the angles a and b. The distance between these two points depends solely on the difference a – b.
It is this geometric observation that underpins the addition formula. We do not start from an arbitrary convention, but from a property of the circle: the distance between two points on the unit circle depends only on the angular difference.
Specifically, the coordinates of M are (cos a, sin a) and those of N are (cos b, sin b). By calculating the square of the distance MN, and then comparing it with the case where one of the angles is zero, we directly obtain cos(a – b) = cos a cos b + sin a sin b.
To delve deeper into the formulas with cos a minus b and sin a, this geometric demonstration remains the most solid starting point.
Did you notice the + sign in the formula for cos(a – b)? It’s counterintuitive. We subtract the angles, but we add the products. This discrepancy between the operation sign on the angles and the sign in the formula is the primary source of errors in exams.
Formula for cos(a – b) and transition to cos(a + b)
Let’s state the complete formula:
cos(a – b) = cos a cos b + sin a sin b
To obtain cos(a + b), replace b with -b. Since cos(-b) = cos b and sin(-b) = -sin b, the sign changes:
cos(a + b) = cos a cos b – sin a sin b
The mnemonic “coco – sisi” (for cos a cos b – sin a sin b) works well for cos(a + b). For cos(a – b), reverse the central sign: “coco + sisi”.
Why this symmetry? Because cosine is an even function. Replacing b with its opposite only modifies the sine, which is odd. The entire mechanics of the addition formulas relies on this parity property of trigonometric functions.
Quick verification with notable angles
Take a = 60° and b = 30°. Then cos(60° – 30°) = cos 30°. The left side equals cos 30°, which is the square root of 3 divided by 2.
The right side: cos 60° cos 30° + sin 60° sin 30°. By substituting the known values, you arrive at the same result. This type of verification takes less than a minute and allows for immediate detection of a sign error.
Sin a and the addition formulas for sine
The function sin a appears in the addition formulas through a similar mechanism. The reference formula is:
sin(a + b) = sin a cos b + cos a sin b
And for the difference:
sin(a – b) = sin a cos b – cos a sin b
Here, the classic mnemonic is “sico + cosi” for sin(a + b). Note that, unlike cosine, the sign in the sine formula follows the sign of the operation on the angles. Addition in the angle, addition in the formula. Subtraction in the angle, subtraction in the formula.
This consistency makes the sine formulas easier to remember than those for cosine. But be careful: the terms cross the functions. In sin(a + b), the first term mixes sin a with cos b, and the second mixes cos a with sin b. This crossing is the second source of frequent errors.
Link between cos(a – b) and sin(a + b)
You can deduce sin(a + b) directly from cos(a – b) using the complementary relationship: sin x = cos(π/2 – x). Apply this identity to sin(a + b) by setting x = a + b:
sin(a + b) = cos(π/2 – a – b) = cos((π/2 – a) – b)
Expand using the formula for cos(a – b), and you arrive at sin a cos b + cos a sin b. All addition formulas can be derived from a single one, that of cos(a – b). This is the formula that should be prioritized for understanding.
Method to remember these formulas in an exam situation
The non-calculator sections of the mathematics baccalaureate now test the ability to mobilize these formulas as automatons. Here’s what works to fix them permanently:
- Memorize only cos(a – b) = cos a cos b + sin a sin b. The other three formulas (cos(a + b), sin(a + b), sin(a – b)) can be derived from it through parity or complementarity.
- Systematically verify with a pair of notable angles (30° and 60°, or 45° and 45°). If the numerical result doesn’t match, it’s the sign that’s wrong.
- Practice deriving sin(a + b) from cos(a – b) using the relationship sin x = cos(π/2 – x). This transition takes less than thirty seconds once automated.
The classic mistake is to memorize the four formulas separately, without understanding their lineage. Remembering one formula and deducing the others reduces the risk of sign inversion.
Concrete cases where cos(a – b) simplifies a calculation
In physics, the phase difference between two sinusoidal signals translates into a term in cos(a – b). Calculating the interference of two waves involves expanding this expression. The formula is not an abstract school exercise.
In analytical geometry, the dot product of two unit vectors at angles a and b is exactly cos(a – b). It’s the same formula, used in a different context. Recognizing this link between the dot product and the addition formula is a considerable time saver in plane geometry problems.
The duplication formulas (cos 2a, sin 2a) also directly follow: just set b = a in the addition formulas. No need to learn them separately.
Understanding cos(a – b) is like holding the thread that connects the majority of trigonometric identities. One well-understood formula, verified with known angles, and derived through reasoning rather than brute memory, is enough to reconstruct everything else on exam day.