There are twelve entries in this grid and no pattern that spans them. The only reliable procedure is to say which function and which reflection before writing anything, and the figure is built so that doing it that way is the easy route.
Press straight to the third reflection, π + θ, and ask what happens to the tangent. Most of a class will say it negates, because the sine and cosine both do and that feels like the pattern. The readout says unchanged.
Then ask why, and let them find it: the tangent is the ratio, and both parts of the ratio flipped. Two minuses. That is a one-line explanation they can reconstruct in an exam, which "tan has period π" is not until they have seen where it comes from.
Then press through the other three and collect the twelve answers into a grid on the board. Doing it in that order matters: the surprising row first, so the grid is built to explain something rather than to be copied down.
| Question | Answer |
|---|---|
| 1. tan(π + 0.7) | 0.8423, unchanged. |
| 2. cos(π − 0.7) | −0.7648, negated. |
| 3. sin(π/2 − 0.7) | 0.7648, which is cos 0.7. |
| 4. The other solution of tan x = 0.5 | 0.4636 + π = 3.6052. |
| 5. Why tan(π + θ) = tan θ | B. Both parts of the ratio negate. |
Question 3's answer is 0.7648, which is also question 2's answer without the minus sign. That is not a coincidence to apologise for; it is the complementary relationship, and a student who notices it has understood the fourth row.
1 markNaming the function and the reflection.
1 markThe relationship, with its sign.
1 markThe answer or the simplified expression.
On a "show that" question the middle mark is the whole question. Writing "tan(π + x) = tan x" as a stated identity earns it, and the arithmetic afterwards is worth less. Tell them that, because students instinctively rush to the number.
| They wrote | What happened |
|---|---|
| −0.8423 on question 1 | The error this page exists for. Assumed the tangent follows the other two. Point at the third row of the grid. |
| 3.8416 on question 1 | Gave π + 0.7, the angle rather than its tangent. Reading error, but check they know which is being asked for. |
| 0.7648 on question 2 | Did not negate. They have used the sine's row on the cosine, which is the commonest confusion between the first two reflections. |
| 0.6442 on question 2 | Gave sin(π − 0.7). Right row, wrong column. |
| 0.6442 on question 3 | Assumed π/2 − θ leaves the sine alone. It does not; it swaps the two, so the answer is cos 0.7. |
| 2.6779 on question 4 | The second error this page exists for. Used the sine's symmetry on a tangent. Its tangent is −0.5, so substituting back catches it. |
| 5.8195 on question 4 | Used the cosine's symmetry, 2π − x. Also gives −0.5. |
| 6.7468 on question 4 | Added 2π. Correct period for sin and cos, wrong for tan, and outside the interval anyway. |
| A grid with one rule in it | They are trying to find a pattern that spans the table. There is not one, and saying so plainly saves them time. |
"Do I have to learn all twelve?" No, and that is worth saying early. Learn the two coordinate facts, that cosine is the across one and sine is the up one, and then each row is one movement of the point. The tangent column is always the product of the other two columns' signs, so there are really eight entries and four of them follow. A class that knows this stops trying to memorise.
"Which reflection do I use to solve an equation?" The one belonging to the function in the equation. Sine partners with π − x, cosine with 2π − x, tangent adds π. It is worth putting those three on the wall, because it is the single most frequently needed fact in the whole topic and it is the one students get wrong under time pressure.
"Why does the cosine use 2π − x?" Because the cosine is even, so cos(−x) = cos x, and −x and 2π − x are the same angle. The first row of the grid and the cosine's solving rule are the same fact, which is a connection worth drawing because it reduces two things to one.
"Is this just 3.5 again?" Partly, and saying so is honest. 3.5 gave them the π − θ row because the ambiguous case needs it. This sub-topic puts all four rows together, and the new content is the π + θ row and the systematic view. If a class finds it familiar, that is a good sign, and the five minutes saved can go on the tangent row.
| Demonstrate | Store the angle, then put tan(a) and tan(π+a) on consecutive lines: 0.8423 twice. Then sin(a) and sin(π+a) underneath: 0.6442 and −0.6442. Four lines, and the asymmetry between the functions is on the screen rather than in an assertion. |
|---|---|
| Where they stick | Retyping 0.7 each time and wondering why the last decimal moves. Store it once; it also makes the point that the angle is not changing, only the reflection. |
| The check | Table mode with a step of π shows the tangent column repeating down the page while the sine and cosine columns alternate. That is the periods made visible, and it is the fastest route to the π + θ row. |
Radians throughout for this one. The reflections are naturally written with π in them and switching to degrees makes the whole table harder to read.
| Step | What |
|---|---|
| 1 | Recall the unit circle: cos across, sin up, tan the ratio. Everything here comes off that. |
| 2 | The figure on the third reflection, π + θ. Ask about the tangent. Collect the wrong answer. |
| 3 | Two minuses cancel. Let them say it. |
| 4 | Then the other three reflections, each as a movement of the point rather than a formula. |
| 5 | Build the grid on the board, tan column last, as the product of the other two. |
| 6 | Connect each row to something they already use: odd and even, the ambiguous case, tangent equations, arcsin + arccos. |
| 7 | tan x = 0.5 on 0 to 2π. Collect the π − x answers and substitute them back. |
| 8 | The three solving rules on the wall: sine π − x, cosine 2π − x, tangent add π. |
Do not say "in the second quadrant everything is negative except sine". It is true and it is a different statement from this grid, and running the two together is how students end up applying quadrant signs to a reflection question. Quadrants tell you the sign of a value; reflections tell you how two values relate. Keep them in separate sentences.
Do not hand over the twelve-entry grid as a table to copy. A copied grid is twelve facts and will not survive a term. The same grid derived from four movements of one point is four facts, and the derivation takes about as long as the copying.