DEPARTMENT OF MATHEMATICS
Core 3
Topic 6 – Trigonometry 1
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Trigonometry 1
cosec, sec and cot
Identities
State the 3rd letter rule to
recall cosec, sec and cot
Solve simple equations
involving cosec, sec and cot
Solving Trigonometric
Equations
Derive the key identities :
1 + tan2x = sec2x
1 + cot2x = cosec2x
from the C2 Identity
sin2x + cos2x = 1
Prove an identity using other
key identities
Inverse Trig Functions
Solve a quadratic equation by
arranging into terms of only
cosec, sec, or cot
Solve equations involving
arcsin, arccos and arctan
(i.e. sin-1, cos-1 and tan-1)
Solve an above-type equation
with multiple angles
(2x, 3x, 4x - 10, etc)
Solve multiple angle equations
involving cosec, sec and cot
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Trigonometry 1
Onto topic number 6 – Trigonometry 1. This will build on work studied at AS
within C2, and alongside Differentiation, is one of the 2 major topics within
C3.
Attached to this document is the formula sheet for C3 – there will be elements
of this topic that you’ll be required to memorise, whilst there will be things
that you need to know are available on the formula sheet, so you’ll only need
to remember how to apply them.
We’ll start by introducing 3 new trigonometric functions, on top of sin, cos and
tan that we learnt last year.
Cosec, Sec and Cot:
Cosec, Sec and Cot are called the reciprocal functions – we’ll see why below:
𝑐𝑜𝑠𝑒𝑐 𝜃 =
1
sin 𝜃
sec 𝜃 =
1
𝑐𝑜𝑠𝜃
cot 𝜃 =
1
𝑡𝑎𝑛𝜃
Here’s an easy way to remember these –
the 3rd letter of cosec, sec and cot is the same as
the 1st letter of sin, cos and tan
You need to remember these, and it is useful to know their graphs (though this is
not crucial).
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Graphs of the Trig Functions (sin, cos and tan are crucial to know):
We’ll look at solving equations involving cosec, sec and cot.
Notes:
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Solving simple equations with cosec, sec and cot:
Worked Examples:
Solving simple equations:
Solve, for 0 ≤ 𝑥 ≤ 360:
a)
sec 𝑥 = −2.5
Solve, for 0 ≤ 𝜃 ≤ 2𝜋:
b)
𝑐𝑜𝑠𝑒𝑐 𝜃 = 2
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Top Tips:
If a question is given with an interval in degrees, you should give
your answer in degrees.
If a question is given with an interval in radians, you should give
your answer in radians.
Never round early in a problem – rounding should always be your
last step to avoid an error with your accuracy!
Solving equations with multiple angles:
Solve, for 0 < 𝛼 < 180
c)
cot(2𝛼 − 45) = √3
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Notes:
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Solve, for 0 < 𝑥 ≤ 2𝜋
d)
1
𝑐𝑜𝑠𝑒𝑐 𝑥 = −√2
2
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Identities involving Cosec, Sec and Cot:
We learnt at C2 the following identity:
sin2 𝑥 + cos 2 𝑥 = 1
From this identity, we derive 2 more crucial identities:
sin2 𝑥 + cos2 𝑥 = 1
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We need to memorise these identities (or at least memorise how to derive them –
I used to do this at the very start of each exam!)
Let’s have a look at some examples of simplifying expressions by using both these
identities and the reciprocal functions in general:
Worked Examples:
a)
Prove the following identities:
𝑐𝑜𝑠𝑒𝑐 4 𝜃 − cot 4 𝜃 ≡
1+cos2 𝜃
1−cos2 𝜃
You can either start on
the LHS and try to work
through until you get to
the RHS,
or you can start on the
RHS and try to work
through to the LHS.
Typically, LHS to RHS will
drop more easily,
although this is not
always true! If you get
stuck, try the other way.
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b)
sec 2 𝜃 − cos 2 𝜃 ≡ sin2 𝜃 (1 + sec 2 𝜃)
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c)
tan 𝑥+cot 𝑥
sec 𝑥 +𝑐𝑜𝑠𝑒𝑐 𝑥
=
1
sin 𝑥+cos 𝑥
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Notes:
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Solving Trigonometric Equations:
This next topic is almost identical to what we studied at C2 – we will be asked to
solve tricky equations involving different trig functions.
Often, it’ll be a hidden quadratic that you’re expected to spot, though sometimes
it will be a little easier than that, and you simply have to spot that you can
factorise something and it should drop!
Let’s have a peek at some examples to make this clearer:
Worked Examples:
a)
Solve, for 0 < 𝑥 < 360:
3 cot 𝑥 = 2 sin 𝑥
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b)
2 cot 2 𝑥 = 7 cosec 𝑥 − 5
If you have cot2 and cosec:
You need to write it in terms of
cosec2 and cosec – then spot the
hidden quadratic!
General Rule:
The squared2 term is the one
that you can change using an
identity, so that is the one that
you’ll need to change!
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c)
2 tan2 𝑥 + 4 sec 𝑥 + sec 2 𝑥 = 2
Think:
You have two squared2 terms
here, and a sec 𝑥 – is there any
way you could leave this in
terms of just sec, so you could
form a quadratic?
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Notes:
Exam Tip:
They have asked similar questions to these before, also including a multiple angle
(e.g. using 3𝑥 instead of 𝑥), so be aware that you may need to change the interval
beforehand on such a question!
[January 2010 – Solve 𝑐𝑜𝑠𝑒𝑐 2 2𝑥 − cot 2𝑥 = 1 for 0 ≤ 𝑥 ≤ 180]
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The Inverse Trig Functions – arcsin, arccos and arctan:
We’ll briefly touch upon these – you have used these all along, throughout this
topic.
arcsin 𝑥 is the same as sin−1 𝑥
arccos 𝑥 is the same as cos −1 𝑥
arctan 𝑥 is the same as tan−1 𝑥
We’ll do one example from a past paper [Jan 2007]:
8.
(i) Prove that
sec2 x – cosec2 x  tan2 x – cot2 x.
(ii) Given that
y = arccos x, –1  x  1 and 0  y  ,
(a)
express arcsin x in terms of y.
(b)
Hence evaluate arccos x + arcsin x. Give your answer in terms of .
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Notes:
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Trigonometry 1 – Questions
Exercise 1:
(Hints for Q6:
sin2 𝑥 + cos 2 𝑥 = 1 ; forming quadratics and factorising)
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Exercise 2:
Exercise 3:
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Exercise 4:
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Exercise 5:
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Solutions to all exercises:
Exercise 1:
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Exercise 2:
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Exercise 3:
Exercise 4:
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Exercise 5:
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