Going from θ to π + θ negates the sine and the cosine and leaves the tangent exactly where it was. At θ = 0.7 the tangent is 0.8423 before and 0.8423 after. There is no single rule here, and that is the whole sub-topic.
Four reflections, three functions, twelve answers. Press through them and watch which column agrees with which: no two reflections treat the three functions the same way.
Each row is a way of moving the angle. Each column is a function. Every row is different and every column is different, which is why there is nothing to memorise here and everything to read off the circle.
| θ becomes | sin | cos | tan | Why |
|---|---|---|---|---|
| −θ | −sin | cos | −tan | Reflection in the x-axis: the height flips, the across does not. |
| π − θ | sin | −cos | −tan | Reflection in the y-axis: the across flips, the height does not. |
| π + θ | −sin | −cos | tan | Rotation of half a turn: both coordinates flip, so their ratio does not. |
| π/2 − θ | cos | sin | cot | Reflection in the line y = x: the two coordinates swap. |
Read the "why" column and the rest follows. Each row is one movement of the point on the unit circle from 3.5, and each entry is what that movement does to a coordinate. The cosine is the across one and the sine is the up one, and the tangent is their ratio, so the tangent's behaviour is always the product of the other two columns' signs.
That last observation is the useful one. You never have to remember the tan column: two minuses in the π + θ row give a plus, and one minus in each of the first two rows gives a minus. Work out sin and cos, then multiply.
The third row is the one that catches people. π + θ sends the point to the diametrically opposite side of the circle, so both coordinates reverse. A sine of 0.6442 becomes −0.6442 and a cosine of 0.7648 becomes −0.7648. But the tangent is the ratio, and (−0.6442)/(−0.7648) = 0.8423, exactly what it was.
That is the same fact as tan has period π, while sin and cos have period 2π. Adding π is a whole period for the tangent and only half a one for the other two. Press the button to the third reflection and watch the tangent marker not move.
So this sub-topic is not a new technique. It is the index to four things you already use, and seeing them in one table is what stops them being four unrelated tricks.
Two solutions, two different spacings. On 0 to 2π, sin x = 0.5 gives 0.5236 and 2.618, which are reflections about π/2 and add to π. But tan x = 0.5 gives 0.4636 and 3.6052, which are π apart and are not reflections of anything. Both equations have two solutions and the two pairs are related completely differently, which is the practical consequence of the table.
Twelve entries, four keystrokes each. Building the table from the machine takes five minutes and is far more durable than being handed it, because the pattern in the "why" column becomes something you noticed.
When you may use it. Analysis Paper 1 is non-calculator, and simplifying an expression with π − θ in it is Paper 1 work. The machine is for constructing the table, once.
The mark people lose. Using the sine's symmetry on a tangent equation. For tan x = 0.5 the second solution is π + 0.4636 = 3.6052, and a student who writes π − 0.4636 = 2.678 gets a number whose tangent is −0.5. The habit: name the function before you pick the symmetry. Sine partners with π − x, cosine with 2π − x, tangent adds π.
Throughout: θ = 0.7 radians, where sin θ = 0.6442, cos θ = 0.7648 and tan θ = 0.8423.
1. Find tan(π + 0.7), to 4 decimal places.
2. Find cos(π − 0.7), to 4 decimal places.
3. Find sin(π/2 − 0.7), to 4 decimal places.
4. tan x = 0.5 has the solution x = 0.4636 on 0 to 2π. Give the other one, to 4 decimal places.
5. Why is tan(π + θ) equal to tan θ, when both sin and cos change sign?
1 markNaming which function and which reflection.
1 markThe correct relationship, with its sign.
1 markThe answer, or the simplified expression.
On a "show that" question the marks are for the reasoning, so write the relationship down as an identity before using it. "Since tan(π + x) = tan x" earns a mark even if the arithmetic afterwards goes wrong.
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