Topic 3.5 · AA Standard Level

The sine stays and the cosine flips

At 60° the sine is 0.866 and the cosine is 0.5. At 120° the sine is 0.866 again and the cosine is −0.5. One kept its value and the other changed sign, from the same angle. There is no single rule, and that is the whole sub-topic.

θ = 60°
0.500cos θ
0.866sin θ
1.732tan θ
firstquadrant

Drag the angle. The cosine is how far across the point is and the sine is how far up, which is the definition and not a fact to remember separately.

The definition, which is a picture

Draw a circle of radius 1 centred on the origin. Measure an angle θ anticlockwise from the positive x-axis, and mark where it meets the circle. That point is (cos θ, sin θ).

That is the definition, not a consequence of one. Everything else on this page is read off the picture:

  1. Cosine is the across one and sine is the up one. Alphabetical order matches coordinate order, which is the only mnemonic worth having here.
  2. Neither can leave −1 to 1, because the point is on a circle of radius 1. A calculator returning sin θ = 1.2 means an error earlier, not an exotic angle.
  3. tan θ = sin θ / cos θ, which is up over across: the gradient of the line from the origin through the point. That is why it is undefined at 90° and 270°, where the line is vertical and has no gradient.

The third one is worth sitting with. tan θ is not a third independent thing to learn; it is a slope. The guide puts it this way too, with the line through the origin written y = x tan θ. At θ = 45° that is y = x, gradient 1, which is exactly tan 45°. At 60° the gradient is 1.732.

Why there is no single rule for the next quadrant

Here is the thing people get wrong. Having learnt that sin(180° − θ) = sin θ, it is natural to assume the cosine behaves the same way. It does not.

Look at the picture rather than the rule. Going from θ to 180° − θ reflects the point in the y-axis. A reflection in the y-axis leaves the height alone and negates the horizontal position. So:

At θ = 60°: the sine is 0.866 and stays 0.866; the cosine is 0.5 and becomes −0.5. Press the mirror button on the figure and watch which leg moves. One rule for both is the error, and the reflection is what tells you which is which.

The same reasoning handles every quadrant without a single extra rule. Find the sign from the picture, then use the acute angle for the size.

QuadrantAnglecossintan
First0 to 90°+++
Second90 to 180°−+−
Third180 to 270°−−+
Fourth270 to 360°+−−

The tangent column is not a fourth thing to memorise: it is the other two multiplied. Two minuses in the third quadrant make the tangent positive, which is why tan 210° = 0.5774, the same as tan 30°, with no minus sign anywhere. Students expect a third-quadrant answer to be negative and it is not.

The exact values, and the only two triangles behind them

The guide asks for 0, π/6, π/4, π/3, π/2 and their multiples. The multiples are the sign work above, so there are really only five values to know, and they come from two triangles: half a square, and half an equilateral triangle.

θcossintan
0100
π/6, 30°√3/2 = 0.866½ = 0.51/√3 = 0.5774
π/4, 45°1/√2 = 0.70711/√2 = 0.70711
π/3, 60°½ = 0.5√3/2 = 0.866√3 = 1.7321
π/2, 90°01undefined

Read the cosine column down and the sine column up: they are the same four numbers in the opposite order. That is the reflection in θ = 45°, and it halves what there is to remember. The π/4 row is the one where they meet, which is why tan π/4 is exactly 1.

On Paper 1 give the surd. √3/2 earns the mark and 0.866 does not, because 0.866 is a rounding of it. Work exactly and convert only if the question asks for a decimal.

What this is for: the sine rule has two answers

Here is the pay-off, and it is the reason the reflection matters rather than being a curiosity. In a triangle, a = 7, b = 10 and A = 40°. Find the third side.

  1. Sine rule: sin B = (10 sin 40°)/7 = 0.9183.
  2. The calculator gives B = 66.67° and stops. But sin(180° − B) = sin B, so B = 113.33° has the same sine and the calculator never mentions it.
  3. Check both for a positive third angle. 180 − 40 − 66.67 = 73.33°, and 180 − 40 − 113.33 = 26.67°. Both are positive, so both triangles exist.
  4. Third side, acute case: 7 sin 73.33°/sin 40° = 10.43. Obtuse case: 7 sin 26.67°/sin 40° = 4.89.

10.43 and 4.89. They differ by 5.54, which is more than half the larger one, so this is not a question where getting one of them is nearly right. Give one answer and you have given half the marks away.

When does the second triangle exist? Only when the obtuse angle leaves something for the third angle, which needs A + B₂ < 180°. That fails whenever A is itself obtuse, so the ambiguity only ever arises when the given angle is acute and the side opposite it is the shorter of the two. Here A = 40° and a = 7 against b = 10, so both conditions hold.

On the GDC: the unit circle and the second angle

The machine will give you one angle from a sine and never the other, which is the single most expensive thing it does on this sub-topic. The useful calculator work here is checking a sign and confirming a second solution you found yourself.

When you may use it. Analysis Paper 1 is non-calculator, and exact values are Paper 1 work: √3/2, not 0.866. Paper 2 is where the ambiguous case usually appears, and there the machine finds the first angle and you supply the second.

TI-Nspire CX II

  1. Set the angle unit deliberately: doc → Settings → Document Settings → Angle. Analysis assumes Radian unless a degree symbol appears
  2. For the signs, put cos(120°) and cos(60°) on consecutive lines: -0.5 and 0.5. Then sin(120°) and sin(60°): 0.866 both times. Two lines that make the whole point
  3. sin⁻¹(0.9183) gives 66.67 and nothing else. The second angle is 180 − 66.67 and you type it yourself
  4. To see that both are solutions, evaluate sin(113.33°): 0.9183, the same as you started with

Casio fx-CG50

  1. SHIFT MENU SET UP → Angle → Deg or Rad, chosen on purpose rather than left as it was
  2. In Run-Matrix, cos(120) then cos(60), then the same for sine. The sine pair agreeing and the cosine pair not is the demonstration
  3. SHIFT sin for sin⁻¹: sin⁻¹(0.9183) gives 66.67, the acute one only
  4. For the exact values, set Input/Output to Math and sin(60) returns √3/2 rather than a decimal, which is the Paper 1 form

The mark people lose. Reporting the acute angle only. The calculator's inverse sine returns one value by design, because a function cannot return two, and the second solution is mathematics the machine is not being asked for. The habit: every time you take an inverse sine in a triangle, write 180 − your answer beside it and then test whether it fits. On an inverse cosine you do not need to, because the cosine is negative in the second quadrant and the calculator will hand you the obtuse angle itself.

Your turn

1. Write down sin 120°, to 3 decimal places.

2. Write down cos 120°, to 1 decimal place.

3. Write down tan 210°, to 4 decimal places.

4. With a = 7, b = 10 and A = 40°, the acute value of B is 66.67°. Give the other possible value of B, to 2 decimal places.

5. Why is tan 210° positive?

Question 5. Why is the tangent of 210 degrees positive?
Where the marks go

1 markThe acute angle from the inverse function.

1 markThe second angle, 180° minus it.

1 markTesting each against the angle sum, and saying which survive.

1 markBoth answers, or a stated reason for rejecting one.

The third mark is the one almost nobody writes down. A line reading "180 − 40 − 113.33 = 26.67 > 0, so this triangle exists" is cheap and it is worth a mark on its own.

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