Doubling the sine instead of the angle is the error, and here it produces 1.3229. That is not a near miss, it is outside the range of a sine, and a student who learns to notice that has a check they can run on every answer for the rest of the course.
Leave it at cos x = 0.75 and ask which of the three bars cannot possibly be a sine. The dashed ceiling at 1 does the teaching. Most classes will point at the tall one straight away, and then the question to ask is how did we know that without working anything out.
Then drag the slider up towards 0.95 and watch the tall bar drop under the ceiling. At cos x = 0.9 the wrong answer is 0.8718, which is perfectly believable. That is the honest and uncomfortable half: the range check catches this error for most of the range and not all of it, and the crossover is at cos x = √3/2 = 0.866. Say so. A check that works most of the time is worth having and is not a substitute for the identity.
The readout underneath changes from "impossible" to "merely wrong" at that threshold, which is the sentence worth collecting from the class.
| Question | Answer |
|---|---|
| 1. sin x | √7/4 = 0.6614, from sin²x = 7/16. |
| 2. sin 2x | 2(√7/4)(3/4) = 3√7/8 = 0.9922. |
| 3. cos 2x | 2(9/16) − 1 = 1/8 = 0.125. |
| 4. tan x | √7/3 = 0.8819. |
| 5. How you know 1.3229 is wrong | B. A sine is a height on a unit circle, so it cannot exceed 1. |
Insist on the surds in the working and the decimal only at the end, if at all. A student who converts √7/4 to 0.6614 at step one carries a rounding through three more operations, and on Paper 1 has thrown away the form the mark scheme wants.
1 markUsing the Pythagorean identity to get the second ratio.
1 markChoosing the sign from the quadrant, and saying so.
1 markThe correct double angle identity, stated before use.
1 markThe answer in exact form.
The second mark is the one students do not know exists. "x is acute, so sin x > 0" is a six-word line and it is a mark. Ask for it every time, including when the sign is obvious, because the habit is what survives into the obtuse questions where it is not.
| They wrote | What happened |
|---|---|
| 1.3229 for sin 2x | The error this page exists for. 2 sin x rather than sin 2x. Point at the ceiling, not at the identity. |
| 0.4375 for sin x | Stopped at sin²x = 7/16. Very common and worth a cheap habit: if the question asks for a ratio and your answer came from an identity, check whether you took the root. |
| 0.25 for sin x | 1 − 3/4, subtracting the cosine rather than its square. They have the identity as a shape rather than a statement. |
| −0.6614 for sin x | Took the negative root without reading "acute". The size is right, so give the method marks and make the quadrant line non-negotiable. |
| −0.125 for cos 2x | Mixed the two cosine forms: 1 − 2cos²x instead of 2cos²x − 1. The squared ratio and the subtraction have to match, and this is the commonest slip on question 3. |
| −0.4375 for cos 2x | cos²x − 1, which is −sin²x. The 2 in front has gone. |
| 0.4961 | sin x cos x with the 2 dropped, on question 2; or a product where question 4 wanted a quotient. Same number, two different errors, so ask which they did. |
| 1.1339 for tan x | cos x over sin x, upside down. It is also 1/0.8819, so they can check it themselves. |
| 7.9373 for tan x | That is tan 2x. They have answered the next question along, which usually means they worked beyond what was asked. |
"Why are there three versions of cos 2x?" Because the Pythagorean identity lets you trade cos² for 1 − sin² at any point, so the three are one statement wearing different clothes. The practical answer is better: use the one built from whatever the question gave you. Given a cosine, 2cos²x − 1 needs no extra work; given a sine, 1 − 2sin²x does. Choosing well saves a line, and choosing badly costs one.
"Can I just find x?" On Paper 2 you can and it will usually work. Three reasons not to: Paper 1 has no calculator and these questions appear there; the exact form is what the mark scheme wants and a decimal from a rounded angle will not match it; and a question can give you cos x = 3/4 with x obtuse, where the inverse cosine hands you the wrong angle and the identity route does not care. The last one is the argument that lands.
"Is it ever true that sin 2x = 2 sin x?" Yes, and it is a good question. It needs 2 sin x cos x = 2 sin x, so either sin x = 0 or cos x = 1, and both of those mean x is a multiple of π, where both sides are 0. So it is true exactly where everything is zero, which is why it feels true and never helps.
"Does the range check work for cosines too?" Yes, and for every answer in the topic. Anything the question calls a sine or a cosine lives in −1 to 1, with no exceptions, because of the radius. Tangents have no such limit, which is worth saying in the same breath so nobody applies the check where it does not belong.
| Demonstrate | Store the angle rather than retyping a rounded one, then put sin(2X) and 2sin(X) on consecutive lines: 0.9922 and 1.3229. Two lines and the misconception is on the screen in the students' own handwriting-equivalent. Do this before the explanation. |
|---|---|
| Where they stick | Retyping 0.6614 instead of storing the angle, then wondering why their fourth decimal place disagrees with the board. Storing is one keystroke and it removes a whole class of complaint. |
| The check | All three cosine forms on three consecutive lines, all returning 0.125. An identity demonstrated beats an identity asserted, and it takes under a minute. |
This is a sub-topic where the calculator's job is to check, not to solve. Say that at the start, because otherwise the strongest students will find the angle and be annoyed when Paper 1 takes the machine away.
| Step | What |
|---|---|
| 1 | The unit circle again, and Pythagoras on it. Derive cos² + sin² = 1 rather than stating it. |
| 2 | Rearrange it both ways. Dwell on the ± and where the question resolves it. |
| 3 | cos x = 3/4, acute. Get sin x as a surd. Collect √7/4. |
| 4 | Ask for sin 2x BEFORE giving the identity. Collect the 1.3229 answers, do not correct them yet. |
| 5 | The figure. The ceiling. Let the class rule their own answer out. |
| 6 | Now the identity, and 3√7/8. |
| 7 | The three cosine forms, and which to pick from what you were given. |
| 8 | tan x for free. Then the slider up to 0.9 and the honest caveat about the range check. |
Do not say "just learn the identities". They are in the formula booklet. What is not in the booklet is which one to reach for and what sign to take, and that is the entire difficulty. A class that has memorised three forms of cos 2x and cannot choose between them is worse off than one that knows two and derives the third.
Do not let "double angle" go unexamined. The name is the error. Every year someone doubles the ratio because the topic is called double angle, and the phrase invites it. Say "the sine of twice the angle" out loud a few times early on; it is clumsier and it is what the thing actually is.