P3 Chapter 3 :: Trigonometry
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Chapter Overview
2:: ‘Solvey’ questions.
3:: ‘Provey’ questions.
4:: Inverse trig functions and their domains/ranges.
Teacher Note: There is no change in this chapter relative to the old pre-2017 syllabus.
A new member of the trig family…
Original and best. Like the ‘Classic Cola’ of trig functions*.
The latter form is particularly useful for differentiation (see Chp9)
We have a convenient way of representing the reciprocal of the trig functions.
* Actually, I’ll contradict this in the ‘Just For Your Interest’ slides coming up soon.
Reciprocal Trigonometric Functions
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Short for “secant”
Pronounced “sehk” in shortened form or “sea-Kant” in full.
Short for “cosecant”
Short for “cotangent”
In shortened form, rhymes with “pot”.
Just for your interest…
A tangent is a trigonometric function (which inputs an angle and gives you the ratio between the opposite and adjacent sides of a right-angled triangle), but also a line which touches a circle. Are they related?
tangent
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Just as ‘radius’ can refer either to the line itself or its length, we have names for other special lines, which can also refer to their lengths:
A tangent is a line which touches the circle. We’re interested in the length just between the touching point and where it meets the secant.
Sine (sort of) comes from the word for ‘bowstring’. It refers to half the line if we doubled up the arc and connected the two ends.
A secant (shortened to ‘sec’) is a line which cuts the circle. In a trig setting, we’re interested in the length from the circle centre to where it meets the tangent.
Calculations
You have a calculator in A Level exams, but won’t however in STEP, etc. It’s good however to know how to calculate certain values yourself if needed.
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Exercises 3A
Pearson Pure Mathematics 3
Page 48-49
Sketches
It touches here because the reciprocal of 1 is 1.
Reciprocating preserves sign. When we divide by a small positive number, we get a very large positive number.
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Sketches
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Sketches
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Example
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Test Your Understanding
Draw the transformations stage by stage, unless you feel comfortable doing multiple transformations at once.
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Exercises 3B
Pearson Pure Mathematics 3
Page 52-53
Questions in the exam usually come in two flavours: (a) ‘provey’ questions requiring to prove some identity and (b) ‘solvey’ questions.
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b
c
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Test Your Understanding
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Solvey Questions
a
b
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Reciprocate both sides.
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Test Your Understanding
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Exercises 3C
Pearson Pure Mathematics 3
Page 56-57
New Identities
There are just two new identities you need to know:
From C2 you knew:
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Fro Tip: I remember this one by starting with the above, and slapping ‘co’ on front of each trig function.
Examples
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Test Your Understanding
Edexcel C3 June 2013 (R)
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Q
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Exercises 3D
Pearson Pure Mathematics 3
Pages 60-61
Inverse Trig Functions
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Inverse Trig Functions
Note that this graph has asymptotes.
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Evaluating inverse trig functions
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Edexcel C3 Jan 2007
One Final Problem…
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Fewer than 10% of candidates got this part right.
Exercises 3E
Pearson Pure Mathematics 3
Pages 64-65