When you come across an expression like cos a + cos b in an exercise, the classic reflex is to look for a numerical value. However, the real time saver comes from elsewhere: transforming this sum into a product to factor and simplify. Here we detail the concrete method, recurring pitfalls, and situations where this formula makes a difference.
Transforming cos a plus cos b into a product: the mechanics to remember
The sum-to-product transformation formula for cosine is written as follows: cos a + cos b = 2 cos((a+b)/2) cos((a-b)/2). We go from a sum of two cosines to a product of two cosines multiplied by 2.
In practice, the method consists of three steps. First, we calculate the half-sum of the angles (a+b)/2, then the half-difference (a-b)/2. We then inject these two results into the formula. The most common pitfall: forgetting the factor of 2 in front of the product, or reversing half-sum and half-difference.
Let’s take a common case in senior year: calculating cos(5pi/12) + cos(pi/12). The half-sum is (5pi/12 + pi/12)/2 = pi/4. The half-difference is (5pi/12 – pi/12)/2 = pi/6. We then obtain 2 cos(pi/4) cos(pi/6), which is 2 x (sqrt(2))/2 x (sqrt(3))/2 = (sqrt(6))/2. Exact result, without a calculator.
We regularly find the formulas for cos a plus cos b in factoring exercises and proofs of trigonometric identities in high school as well as in preparatory classes.
Trap of the sign between cos a plus cos b and cos a minus cos b
The most costly confusion in tests concerns the transition from sum to difference. For cos a + cos b, we obtain a product of two cosines. For cos a – cos b, the result involves two sines with a negative sign: cos a – cos b = -2 sin((a+b)/2) sin((a-b)/2).
The change of function (cosine to sine) and the appearance of the minus sign are not intuitive. It is recommended to write the two formulas side by side on your scratch paper before starting an exercise. The time savings are significant compared to trying to recall the formula mid-calculation.
A useful mnemonic: the sum of cosines gives a product of cosines (we stay in the same family), while the difference of cosines shifts to a product of sines. This logic of “trigonometric family” helps avoid confusing the two.
Solving a trigonometric equation through factoring
Factoring using the sum-to-product formula becomes particularly useful when encountering an equation of the form cos a + cos b = 0. Rather than looking for solutions angle by angle, we factor: 2 cos((a+b)/2) cos((a-b)/2) = 0.
A zero product implies that at least one of the factors is zero. We end up with two simple equations to solve separately:
- cos((a+b)/2) = 0, which gives (a+b)/2 = pi/2 + k pi, or a+b = pi + 2k pi
- cos((a-b)/2) = 0, which gives (a-b)/2 = pi/2 + k pi, or a-b = pi + 2k pi
This decomposition is faster and more reliable than manipulating the sum in its original form. It avoids sign errors and allows us to handle each case independently.
Degrees or radians: a source of silent error
The formula cos a + cos b = 2 cos((a+b)/2) cos((a-b)/2) works equally in degrees and radians, as long as you do not mix the two units in the same calculation. In practice, high school statements use radians, but errors frequently occur when one angle is given in degrees and the other in radians.
Before applying the formula, check that both angles are expressed in the same unit. If one is 60 degrees and the other is pi/4, convert first. This check takes a few seconds and avoids an absurd result.
Addition formulas and sum-to-product formula: do not confuse
Two families of trigonometric formulas are often mixed up. The addition formulas give the cosine or sine of a sum of angles:
- cos(a + b) = cos a cos b – sin a sin b
- cos(a – b) = cos a cos b + sin a sin b
- sin(a + b) = sin a cos b + cos a sin b
The sum-to-product formulas, on the other hand, transform a sum of trigonometric functions (cos a + cos b) into a product. The two families are related but serve different needs. The addition formulas are used to expand cos(a+b), while the sum-to-product formula is used to factor cos a + cos b.
In preparatory classes, the sum-to-product formula is actually derived from the addition formulas. We add cos(a+b) and cos(a-b), obtaining 2 cos a cos b, then we make a change of variable to retrieve the classic form. This logical connection between the two families helps reconstruct the formula if it is forgotten on exam day.
To distinguish the two in an exercise situation, we look at the starting form. If we have cos(a+b), we expand using the addition formulas. If we have cos a + cos b (two terms separated by a plus), we factor using the sum-to-product formula. The form of the expression dictates the formula to use.
Keeping these sorting reflexes allows us to approach trigonometry exercises without hesitation, whether for a senior year math specialty assignment or a preparatory class oral exam.
