Trig functions are not linear: $\sin(a+b)$ is not $\sin a + \sin b$, and $\sin 2x$ is not $2\sin x$

01The mistake

Students write $\sin(a+b) = \sin a + \sin b$ and $\sin(2x) = 2\sin x$. Both are false for essentially every input. The second is the more common on the test, because $2x$ looks like a coefficient sitting inside and coefficients are things students pull out.

The tell is that the angle sum identities seem unmotivated. If $\sin(a+b)$ could simply be split, there would be no reason for $\sin a \cos b + \cos a \sin b$ to exist. A student who cannot say why the identity is needed has not registered that the naive version fails, and will keep reverting to it under pressure.

It generalises across the whole course: $\log(a+b) \neq \log a + \log b$, $\sqrt{a+b} \neq \sqrt a + \sqrt b$, $f(a+b) \neq f(a)+f(b)$. Students often hold all of these simultaneously and treat them as separate topics, which is why correcting one rarely fixes the others — the belief being corrected is more general than the example.

Watch for the logarithm case in particular, because it is genuinely confusing: $\log(ab) = \log a + \log b$ is true. A student who has learned that rule and not its precise form will happily produce $\log(a+b) = \log a + \log b$, having remembered that logs turn something into a sum.

02Why it makes sense to the student

The notation looks like multiplication. $\sin(a+b)$ has a symbol immediately followed by a parenthesised expression, which is exactly how $3(a+b)$ is written. In one case distribution is legal and in the other the symbol is not a factor at all, and nothing visible distinguishes them.

Students often do not read $\sin$ as a function name. Asked what $\sin$ is, many describe it as a button, an operation, or a value — not as a rule that takes an input to an output. Without the function concept there is no reason to expect it to behave differently from a coefficient.

Distribution is the most rewarded move in their algebra. When in doubt, expand: it works for products over sums, and it is what a student under time pressure reaches for by default.

And the identities are usually taught as things to memorise for later use rather than as answers to a question. If the angle sum formula is presented as a fact to be stored, students never encounter the failure it exists to repair, so nothing displaces the naive version.

03The correction

Test it numerically, immediately, and let the numbers settle it. $\sin(30° + 60°) = \sin 90° = 1$. But $\sin 30° + \sin 60° = 0.5 + 0.866 = 1.366$. Not equal, not a rounding issue. Do the same for $\sin(2 \cdot 30°) = \sin 60° = 0.866$ against $2\sin 30° = 1$. Two counterexamples take a minute and are more durable than any rule.

Then say what $\sin$ actually is: a function name. $\sin(x)$ means “the sine of $x$,” the output of a rule applied to the input $x$. It is not $\sin$ multiplied by $x$, so there is nothing to distribute. Writing it as $\sin(x)$ with explicit brackets, rather than $\sin x$, helps more than it looks like it should.

Present the angle sum identity as the repair rather than as a fact. Since $\sin(a+b)$ cannot be split naively, we need to know what it does equal — and that is $\sin a \cos b + \cos a \sin b$. Taught in that order, the identity has a job. Taught in the other order, it is one more formula.

Name the general principle so it transfers: functions do not distribute over addition. Sine, cosine, logarithm, square root, squaring — none of them. Then handle the logarithm rule precisely, because it is the exception students misremember: $\log(ab) = \log a + \log b$ converts a product to a sum. It says nothing about $\log(a+b)$, which does not simplify at all.

A useful classroom test: put $\sin(a+b)$, $\log(ab)$, $\log(a+b)$, $\sqrt{ab}$ and $\sqrt{a+b}$ on the board and ask which can be split into two terms. Only $\log(ab)$ can, and $\sqrt{ab} = \sqrt a \sqrt b$ splits into a product rather than a sum. Students who split all five have the general belief, and the ones who split exactly the right two have the concept.

04A sample question

05What each wrong answer reveals

• A The coefficient pulled out. The student has treated $\sin$ as a factor and 2 as something that can be moved past it. Test at $x = 30°$: $\sin 60° = 0.866$ while $2\sin 30° = 1$. This is the misconception in its most common form and the numerical check is the correction — a student who has computed both values once tends not to do it again.
• B Correct. The double angle identity gives $\sin(2x) = 2\sin x\cos x$. It follows from the angle sum identity with $a = b = x$, which is worth showing rather than asserting.
• C The 2 read as an exponent. The student has moved the 2 from the argument to a power on the function, conflating $\sin(2x)$ with $\sin^2 x$ — two expressions where the same digit means completely different things. This is a notation-parsing failure and points at the $\sin^2$ versus $\sin^{-1}$ ambiguity, so it often travels with U2-PR6.
• D The same error, written additively. $\sin x + \sin x$ is $2\sin x$, so this is A in another form. Worth noting when a student picks D over A, because it usually means they expanded $2x$ as $x + x$ inside the argument and then distributed — showing the distribution step explicitly rather than performing it silently. Same repair, and the student has helpfully shown their reasoning.

A and D are one misconception in two notations, and a class split across both is more uniform than it appears. C is a different problem — the superscript and the argument are being confused — and it will resurface at $\sin^{-1}$ rather than at the angle sum identities. The numerical counterexample fixes A and D; C needs the notation read aloud.

06Try it in Mistake Master