While going through a trigonometry worksheet, you may have searched for an identity for sin a + cos b. And you found nothing. No boxed formula, no associated proof. This absence has a precise explanation: sin a + cos b is not a trigonometric identity, it is a sum of two terms that depend on different angles and different functions. Understanding why this expression cannot be simplified is as educational as mastering the classic addition formulas.
Why sin a + cos b cannot be simplified like sin(a+b)
The addition formulas relate sin(a+b) or cos(a+b) to the sines and cosines of a and b taken separately. For example, sin(a+b) = sin a cos b + cos a sin b. Each term on the right side mixes the two angles a and b through products.
With sin a + cos b, the situation is different. You are adding the sine of one angle and the cosine of another, without any cross product. The two terms each exist on their own axis of the trigonometric circle: sin a is the vertical coordinate of the point associated with angle a, while cos b is the horizontal coordinate of the point associated with angle b.
To delve deeper into this point, you can check the formula sin a plus cos b on Kömal, which details this distinction with diagrams.
No standard identity allows merging these two terms into one. When you see sin a + cos b in an exercise, the right reaction is not to try to simplify, but to work with each term separately or to transform one of the two to harmonize the expression.
Transforming cos b into sine to harmonize the expression
You may have noticed that cos and sin are the same curve, just shifted? The relation cos b = sin(pi/2 – b) allows us to rewrite the expression in a homogeneous form:
sin a + cos b = sin a + sin(pi/2 – b)
Now, both terms are sines. We can apply the sum-to-product transformation formula:
sin p + sin q = 2 sin((p+q)/2) cos((p-q)/2)
By setting p = a and q = pi/2 – b, we obtain:
sin a + cos b = 2 sin((a + pi/2 – b)/2) cos((a – pi/2 + b)/2)
This factored form is not a “magic simplification.” It remains longer than the original expression. Its usefulness appears in specific contexts:
- Solving an equation of the type sin a + cos b = 0, since a zero product implies that at least one factor is zero
- Studying the sign of the expression over a given interval by analyzing each factor separately
- Calculating certain integrals where the product form lends itself better to calculation techniques
Outside of these cases, keeping sin a + cos b as is often remains the most readable choice.
Special case: sin a + cos a with a single angle
When the two angles are identical (a = b), the situation changes. The expression sin a + cos a no longer mixes two distinct angles, and a useful factorization exists.
We use the “reverse” addition formula. The goal is to write sin a + cos a in the form R sin(a + phi), where R is an amplitude and phi is a phase shift.
sin a + cos a = square root of 2 times sin(a + pi/4)
How to derive this result
We start from the formula sin(a + pi/4) = sin a cos(pi/4) + cos a sin(pi/4). The values cos(pi/4) and sin(pi/4) both equal square root of 2 divided by 2. By multiplying by square root of 2, we exactly reconstruct sin a + cos a.
This result is widely used in senior year and preparatory classes. It allows for direct reading of the amplitude (square root of 2) and the phase shift (pi/4) of a sinusoidal signal. Any linear combination of sin and cos of the same angle reduces to a single sine with amplitude and phase.
Recognizing the right situation in a trigonometry exercise
When faced with a trigonometric expression, the first question to ask is: do the two terms refer to the same angle or to different angles?
- Same angle (sin a + cos a, 3 sin a – 2 cos a): factorization in the form R sin(a + phi) or R cos(a + psi) is always possible
- Different angles (sin a + cos b): no direct identity, convert to identical functions then sum-to-product formula if necessary
- Sum of the same function (sin a + sin b, cos a + cos b): direct factorization formula, without prior conversion
This framework avoids wasting time searching for a formula that does not exist. In middle school, trigonometry remains linked to the right triangle (side ratios). The addition and factorization formulas appear in the curriculum of the first and final years, and are then deepened in preparatory classes.
Common trap in exams
Writing sin a + cos b = sin(a+b) is a classic mistake. The right side expands to sin a cos b + cos a sin b, which only gives sin a + cos b in very particular cases (cos b = 1 and cos a sin b = 0, for example). In the general case, this equality is false.
Trigonometry rewards method more than memory. Knowing that an expression cannot be simplified is already useful information: it guides towards the right resolution strategy instead of going in circles over a formula.
