At x = 0.5, sin⁻¹x is 0.5236 and 1/sin x is 2.0858. The notation looks like an index and is not one, and the two functions are not even defined in the same places.
The dashed box is where arcsin lives. Cosec never enters it, in either direction. Two functions that share no point of the plane are not easy to confuse once you have seen them together.
These are just names for three fractions:
| Name | Means | At θ = 1 |
|---|---|---|
| sec θ | 1 / cos θ | 1/0.5403 = 1.8508 |
| cosec θ | 1 / sin θ | 1/0.8415 = 1.1884 |
| cot θ | 1 / tan θ | 1/1.5574 = 0.6421 |
sec goes with cos and cosec goes with sin. That crossover is the thing people get wrong, and it is worth noticing that the names are deliberately mismatched: the co is on the opposite one each time. No mnemonic for this is reliable, so do the thing that is: write "sec = 1/cos" at the top of the page before you start, every time, and read it off rather than recalling it.
None of the three is ever between −1 and 1, excluding zero, because each is one divided by something that never exceeds 1 in size. So sec θ = 0.5 has no solutions, which is the range check for this sub-topic and the mirror image of the one at 3.6.
Start from cos²θ + sin²θ = 1, which is 3.6, and divide it by cos²θ:
1 + tan²θ = sec²θ
Then divide the original by sin²θ instead:
cot²θ + 1 = cosec²θ
That is the whole derivation, and it is worth doing rather than learning, because the division tells you which identity you are getting: divide by cos² and you get the sec one; divide by sin² and you get the cosec one.
Check at θ = 1: tan 1 = 1.5574, so 1 + tan² = 3.4255; and sec 1 = 1.8508, so sec² = 3.4255. The same to every decimal place the calculator will show.
Use them in the same way as the original: to trade one ratio for another when a question gives you the wrong one. A question with both tan and sec in it is asking for the first identity, and the substitution usually turns the problem into a quadratic.
sin, cos and tan are not one-to-one, so strictly they have no inverses. The course fixes this the same way 2.14 fixed x²: restrict until it is one-to-one, then invert. The restriction chosen is what gives each inverse its range.
| Function | Domain | Range |
|---|---|---|
| arcsin x | −1 ≤ x ≤ 1 | −π/2 to π/2, which is −1.571 to 1.571 |
| arccos x | −1 ≤ x ≤ 1 | 0 to π, which is 0 to 3.142 |
| arctan x | every real x | −π/2 to π/2, not reached |
Read the domain column first. arcsin 1.2 does not exist, because no angle has a sine of 1.2, and a calculator will error rather than guess. arctan is the exception: it accepts anything, because a tangent can be any size. arctan 1000 = 1.5698, creeping towards π/2 = 1.5708 without ever arriving, which is the horizontal asymptote of its graph.
The range column is why your calculator gives one answer when a triangle has two. arcsin returns something in −π/2 to π/2 by definition, so it can never hand you an obtuse angle. That is the 3.5 ambiguous case, explained: the machine is not being unhelpful, it is returning the only value its range allows.
One neat relation worth knowing. arcsin x + arccos x = π/2, for every x in −1 to 1. At x = 0.5 that is 0.5236 + 1.0472 = 1.5708. It follows from sin(π/2 − θ) = cos θ, and it makes a pleasant check: if your two inverse values do not add to 1.5708, one of them is wrong.
The machine has no sec, cosec or cot key, which surprises people. You type the reciprocal yourself, and that is the clearest possible statement of what these ratios are.
When you may use it. Analysis Paper 1 is non-calculator, and identity work is Paper 1. The useful calculator job here is proving to yourself that sin⁻¹ and 1/sin are different, which takes two lines.
The mark people lose. Writing sin⁻¹x for 1/sin x, or reading a question's sin⁻¹ as a reciprocal. The notation is genuinely unfortunate and the examiners know it, which is why they write arcsin in the guide. The habit: read sin⁻¹ aloud as "arcsin", never as "sin to the minus one". If you say the word you cannot make the error.
1. Find arcsin 0.5, in radians to 4 decimal places.
2. Find sec 1, to 4 decimal places. Work in radians.
3. Given tan θ = 1.5574, find sec²θ to 4 decimal places.
4. Find arcsin 0.5 + arccos 0.5, to 4 decimal places.
5. Why does sec θ = 0.5 have no solutions?
1 markThe correct identity quoted before use.
1 markThe substitution, with the right ratio traded for the right one.
1 markThe answer, with any impossible value rejected and the reason given.
The rejection mark is the same one as at 3.8. A quadratic in sec θ will often hand you a root between −1 and 1, and "no solutions since |sec θ| ≥ 1" is the line that earns it.
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