sin(45° + 45°) is 1. sin 45° + sin 45° is 1.4142, which is not a sine of anything. The identities exist because the obvious move is not available.
B is fixed at 45°. The two bars are equal at A = 0, because sin 0 is zero, and that is the only place. Everywhere else the wrong answer is too big.
These are in the formula booklet, so the work is in using them, not in recalling them:
| Identity | The sign to watch |
|---|---|
| sin(A ± B) = sin A cos B ± cos A sin B | The sign follows the one in the bracket. |
| cos(A ± B) = cos A cos B ∓ sin A sin B | The sign flips. Plus in the bracket gives minus in the middle. |
| tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B) | Follows on top, flips underneath. |
The cosine one is the one that gets written wrong. cos(A + B) has a minus in it, and there is a reason worth knowing: at A = B = 45° the left side is cos 90° = 0, and cos45cos45 + sin45sin45 would be 0.5 + 0.5 = 1. Only the minus version gives zero. One substitution checks which sign you wrote, and A = B = 45° is the cheapest one to use.
Why sin(A + B) cannot be sin A + sin B. Move the slider and watch the warm bar pass the ceiling. From A = 17.03° onwards the wrong answer is above 1, so it is not merely inaccurate: it is not a value a sine can take.
And notice where the two agree: only at A = 0, where sin A is zero and adding it does nothing. That is the general pattern for a false distribution law. It works where one term vanishes, which is often enough for the habit to form.
75° is not one of the angles with an exact value, but 45 + 30 is, and both of those are. That is what the identities are for.
sin 75° = sin(45° + 30°) = sin45cos30 + cos45sin30
= (1/√2)(√3/2) + (1/√2)(1/2) = √3/(2√2) + 1/(2√2)
= (√3 + 1)/(2√2) = (√6 + √2)/4
which is 0.965926. The last step rationalises the denominator, and the form (√6 + √2)/4 is what a mark scheme will want.
The same two angles give cos 75° = (√6 − √2)/4 = 0.258819, which is also sin 15°, and tan 75° = 2 + √3 = 3.732051.
Compare that with the wrong answer here: sin45 + sin30 = 1.2071, which exceeds 1 by 0.2071. The range check catches it instantly, and on Paper 1 it is the only check available.
The guide asks for this one to come from the compound angle identity, which is three lines. Put B = A in the tangent identity:
tan(A + A) = (tan A + tan A)/(1 − tan A tan A)
tan 2A = 2 tan A / (1 − tan²A)
Check at A = 30°: tan 30 = 0.5774, so 2(0.5774)/(1 − 0.3333) = 1.1547/0.6667 = 1.732051, which is tan 60° exactly. The naive answer, 2 tan 30 = 1.1547, is the numerator on its own.
The same substitution gives the two you already have from 3.6: sin 2A = 2 sin A cos A, and cos 2A = cos²A − sin²A. All three double angle identities are one substitution away from the compound ones, which is three fewer things to carry.
A neat consequence: at A = 45° the denominator becomes 1 − 1 = 0, so tan 90° comes out undefined. The formula knows, which is a small sign that it is the right formula.
The risk on this sub-topic is a sign, and a sign error gives a plausible number rather than an error. One substitution into both sides catches it, and the machine does that in two lines.
When you may use it. Analysis Paper 1 is non-calculator, and exact values like (√6 + √2)/4 are Paper 1 answers. Use the machine to verify your identity while learning, then work exactly.
The mark people lose. The sign in cos(A + B). It flips, and nothing about the expression reminds you. The habit: substitute A = B = 45° into whatever you have written. cos 90° is 0, so the right version gives 0 and the wrong one gives 1. It takes five seconds and it is the only check that works without a calculator.
1. Find sin 75°, to 6 decimal places.
2. Find cos 75°, to 6 decimal places.
3. Using tan 30° = 0.577350, find tan 60° from the double angle identity, to 6 decimal places.
4. Find sin 45° + sin 30°, to 6 decimal places. This is the wrong answer to question 1, and it is worth seeing the number.
5. Which substitution tells you fastest whether you have written cos(A + B) with the right sign?
1 markSplitting the angle into two with known exact values.
1 markThe identity quoted, with the right signs.
1 markSubstituting the exact surd values.
1 markThe answer simplified, with a rationalised denominator.
The first mark is cheap and often skipped: writing "75 = 45 + 30" earns it on its own. The last one is the one students leave out, because (√3 + 1)/(2√2) looks finished and is not.
These pages are free and stay free, but they are general and your IA is not. Send me your research question, or whatever exists so far, and I will tell you in writing whether the topic has a ceiling on it, where the marks are going, and what to change first. That costs nothing and it comes back within 24 hours.
Written by a serving IB Diploma and Career-related Programme Coordinator and Head of Mathematics, who reads internal assessments across every subject group every year. If you then want the whole draft reviewed properly against all five criteria, that is the paid one, and it is refunded if it does not name at least three specific things to fix.
Already have a full draft? Have the whole thing reviewed against all five criteria, $99.