The identities are in the booklet, so the only thing that can go wrong is a sign, and the cosine sign flips. One substitution settles it in five seconds and works without a calculator, which is the whole point on Paper 1.
Set the slider to 0 first. The two bars are the same height, 0.707 each, and somebody will say the rule works. Then move it to 30 and the warm bar goes through the ceiling.
The number to collect is 17.03°, which the figure's caption line reports as the switch from "merely wrong" to "impossible". Below that the wrong answer is still a legal sine and only the mathematics catches it; above it, the range check does. Say both halves, because a class told only about the ceiling will assume the check always works.
Then slide to 135°, where the true value is 0 and the wrong one is 1.414. The two are as far apart as they get in that window, and a rule that can be out by 1.414 on a quantity bounded by 1 is not a near miss.
| Question | Answer |
|---|---|
| 1. sin 75° | (√6 + √2)/4 = 0.965926. |
| 2. cos 75° | (√6 − √2)/4 = 0.258819. |
| 3. tan 60° from the double angle | 1.732051, which is √3. |
| 4. sin 45° + sin 30° | 1.207107, the wrong answer, deliberately. |
| 5. The sign check | B. A = B = 45°, because cos 90° = 0. |
Question 4 asks for a wrong answer on purpose, and it is worth saying why in class: a student who has computed 1.207107 and seen it is above 1 will not produce it by accident later. Naming an error numerically fixes it better than warning against it.
1 markSplitting the angle into two with known exact values.
1 markThe identity quoted with correct signs.
1 markSubstituting the exact surds.
1 markSimplifying, with the denominator rationalised.
Writing "75 = 45 + 30" earns the first mark on its own and takes four characters. The last one is the commonly dropped one: (√3 + 1)/(2√2) is a correct value and not a finished answer, and students who stop there lose a mark on work that was entirely right.
| They wrote | What happened |
|---|---|
| 1.207107 for sin 75° | The error this page exists for. Distributed the sine. Above 1, so the range check catches it without the identity. |
| 0.965926 for cos 75° | The sign. They used plus, which gives cos(45 − 30) = cos 15°. Awkwardly, that equals sin 75°, so it looks like the previous answer and feels plausible. Worth flagging explicitly. |
| 0.258819 for sin 75° | Either the cosine identity used by mistake, or a minus where the bracket said plus. Ask which identity they wrote down. |
| 1.154701 on question 3 | 2 tan 30°, the numerator alone. The denominator 1 − tan² is the half that makes it an identity rather than a guess. |
| 0.333333 on question 3 | tan²30°. They have computed a piece and stopped. |
| 0.707107 anywhere | One term of the expansion on its own, usually because they wrote the identity and then substituted into half of it. |
| (√3 + 1)/(2√2) as a final answer | Correct but unsimplified. Give the method marks, withhold the last one, and show the rationalising step once. |
| A decimal on Paper 1 | 0.966 earns nothing where the exact form exists. The question will have been built from 45 and 30 precisely so that it does. |
"Which two angles do I split it into?" Whichever two from the exact list add or subtract to give it. 75 = 45 + 30, and 15 = 45 − 30. 105 = 60 + 45. The list is 0, 30, 45, 60, 90 and their multiples, so the reachable angles are the sums and differences of those, and in practice an exam will give you one that works.
"Why does the cosine sign flip?" The honest short answer is that it falls out of the derivation, and the derivation is not on the course. The useful answer is the substitution: at A = B = 45° the left side is 0, and only the minus version produces 0. Teach the check rather than the reason, because the check is what they need in an exam and it is reliable.
"Can I use these to get sin 2A?" Yes, and the guide asks for exactly that: put B = A. All three double angle identities come out in one line each, which is worth doing in front of them, because it turns six identities to remember into three plus a substitution.
"Is tan(A + B) in the booklet?" Yes, and so is the double angle for tan, but students often hunt for them because they look unfamiliar. Point them out explicitly: the time lost looking is worse than the time spent learning them.
| Demonstrate | Put cos(45+45), then the minus version, then the plus version, on three consecutive lines: 0, 0, 1. The sign test, performed rather than asserted, and it takes under a minute. Do it before you tell them which sign is right. |
|---|---|
| Where they stick | Degrees and radians. These examples are in degrees, the rest of the topic is in radians, and a class with the machines set inconsistently will produce numbers nobody can reconcile. Set it together and say you are switching back afterwards. |
| The check | Six decimal places, not two. sin(75) and the expansion both give 0.965926; a sign error elsewhere in the topic can agree to two places and not to six. |
Also show (√(6)+√(2))/4 returning the same decimal. Students who have done the algebra correctly but distrust their surd benefit from seeing the three routes agree.
| Step | What |
|---|---|
| 1 | Ask for sin 75° with no calculator and no identities. Collect the guesses, including 1.207. |
| 2 | Ask whether 1.207 can be a sine. The range check kills it before any identity appears. |
| 3 | The figure, slider from 0 upwards. Agreement only at 0, impossible from 17.03°. |
| 4 | The four identities from the booklet. Spend the time on the two signs, not on the forms. |
| 5 | The A = B = 45° check, done on the calculator. |
| 6 | sin 75° worked exactly, including the rationalising step. |
| 7 | cos 75° and tan 75° from the same split. |
| 8 | B = A, and all three double angle identities derived in three lines. |
Do not say "the sign changes for cos". It is too short to be useful and students cannot reconstruct which way. Say "plus in the bracket, minus in the middle", and then give them the 45 and 45 check so they never have to trust the phrasing.
Do not teach the double angle identities separately from these. The guide is explicit that they should be derived from the compound ones, and a class that learns them as a separate list carries six facts where three plus a substitution would do. It also loses the connection that makes tan 2A memorable.