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P2 Chapter 7 :: Trigonometry And Modelling
@DrFrostMaths
Last modified: 23rd November 2017
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Chapter Overview
sin ๐ + ๐ = sin ๐ cos ๐ + cos๐ sin ๐
1a:: Addition Formulae
โSolve, for 0 โค ๐ฅ < 2๐, the equation2 tan 2๐ฆ tan ๐ฆ = 3
giving your solutions to 3sf.โ
1b:: Double Angle Formulae
โFind the maximum value of 2 sin ๐ฅ + cos ๐ฅ and the value of ๐ฅfor which this maximum occurs.โ
3:: Simplifying ๐ cos ๐ฅ ยฑ ๐ sin ๐ฅ
This chapter is very similar to the trigonometry chapters in Year 1. The only difference is that new trig functions: ๐ ๐๐, ๐๐๐ ๐๐ and ๐๐๐ก, are introduced.
โThe sea depth of the tide at a beach can be
modelled by ๐ฅ = ๐ ๐ ๐๐2๐๐ก
5+ ๐ผ , where ๐ก is
the hours after midnight...โ
4:: Modelling
Teacher Note: There is again no change in this chapter relative to the old pre-2017 syllabus.
Addition Formulae
Addition Formulae allow us to deal with a sum or difference of angles.
๐ฌ๐ข๐ง ๐จ + ๐ฉ = ๐ฌ๐ข๐ง๐จ ๐๐จ๐ฌ๐ฉ + ๐๐จ๐ฌ๐จ ๐ฌ๐ข๐ง๐ฉ๐ฌ๐ข๐ง ๐จ โ ๐ฉ = ๐๐๐๐จ๐๐๐๐ฉ โ ๐๐๐๐จ๐๐๐๐ฉ
๐๐จ๐ฌ ๐จ + ๐ฉ = ๐๐จ๐ฌ๐จ ๐๐จ๐ฌ๐ฉ โ ๐ฌ๐ข๐ง๐จ ๐ฌ๐ข๐ง๐ฉ๐๐จ๐ฌ ๐จ โ ๐ฉ = ๐๐จ๐ฌ๐จ ๐๐จ๐ฌ๐ฉ + ๐ฌ๐ข๐ง๐จ ๐ฌ๐ข๐ง๐ฉ
๐ญ๐๐ง ๐จ + ๐ฉ =๐ญ๐๐ง๐จ + ๐ญ๐๐ง๐ฉ
๐ โ ๐ญ๐๐ง๐จ ๐ญ๐๐ง๐ฉ
๐๐๐ ๐จ โ ๐ฉ =๐๐๐๐จ โ ๐๐๐๐ฉ
๐ + ๐๐๐๐จ ๐๐๐๐ฉ
Do I need to memorise these?Theyโre all technically in the formula booklet, but you REALLY want to eventually memorise these (particularly the ๐ ๐๐ and ๐๐๐ ones).
How to memorise:
First notice that for all of these the first thing on the RHS is the same as the first thing on the LHS!
โข For sin, the operator in the middle is the same as on the LHS.
โข For cos, itโs the opposite.โข For tan, itโs the same in
the numerator, opposite in the denominator.
โข For sin, we mix sin and cos.
โข For cos, we keep the cosโs and sinโs together.
Common Schoolboy/Schoolgirl Error
Why is sin(๐ด + ๐ต) not just sin ๐ด + sin(๐ต)?Because ๐๐๐ is a function, not a quantity that can be expanded out like this. Itโs a bit like how ๐ + ๐ ๐ โข ๐๐ + ๐๐.We can easily disprove it with a counterexample.
?
Addition Formulae
Now can you reproduce them without peeking at your notes?
๐ฌ๐ข๐ง ๐จ + ๐ฉ โก ๐ฌ๐ข๐ง๐จ ๐๐จ๐ฌ๐ฉ + ๐๐จ๐ฌ๐จ ๐ฌ๐ข๐ง๐ฉ๐ฌ๐ข๐ง ๐จ โ ๐ฉ โก๐๐๐๐จ๐๐๐๐ฉ โ ๐๐๐๐จ๐๐๐๐ฉ
๐๐จ๐ฌ ๐จ + ๐ฉ โก๐๐จ๐ฌ๐จ ๐๐จ๐ฌ๐ฉ โ ๐ฌ๐ข๐ง๐จ ๐ฌ๐ข๐ง๐ฉ๐๐จ๐ฌ ๐จ โ ๐ฉ โก ๐๐จ๐ฌ๐จ ๐๐จ๐ฌ๐ฉ + ๐ฌ๐ข๐ง๐จ ๐ฌ๐ข๐ง๐ฉ
๐ญ๐๐ง ๐จ + ๐ฉ โก๐ญ๐๐ง๐จ + ๐ญ๐๐ง๐ฉ
๐ โ ๐ญ๐๐ง๐จ ๐ญ๐๐ง๐ฉ
๐๐๐ ๐จ โ ๐ฉ โก๐๐๐๐จ โ ๐๐๐๐ฉ
๐ + ๐๐๐๐จ ๐๐๐๐ฉ
How to memorise:
First notice that for all of these the first thing on the RHS is the same as the first thing on the LHS!
โข For sin, the operator in the middle is the same as on the LHS.
โข For cos, itโs the opposite.โข For tan, itโs the same in
the numerator, opposite in the denominator.
โข For sin, we mix sin and cos.
โข For cos, we keep the cosโs and sinโs together.
??
??
?
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Proof of sin ๐ด + ๐ต โก sin ๐ด cos๐ต + cos๐ด sin ๐ต
๐ต
๐ด
1
(Not needed for exam)
1: Suppose we had a line of length 1 projected an angle of ๐ด + ๐ต above the horizontal.Then the length of ๐๐ =sin ๐ด + ๐ตIt would seem sensible to try and find this same length in terms of ๐ด and ๐ต individually.
๐ ๐
๐
sin(๐ด
+๐ต)
2: We can achieve this by forming two right-angled triangles.
๐ด
3: Then weโre looking for the combined length of these two lines.
4: We can get the lengths of the top triangleโฆ
sin๐ดcos๐ต
cos๐ดsin๐ต
5: Which in turn allows us to find the green and blue lengths.
6: Hence sin ๐ด + ๐ต =sin๐ด cos๐ต + cos๐ด sin๐ต
โก
Proof of other identities
Can you think how to use our geometrically proven result sin ๐ด + ๐ต = sin๐ด cos๐ต + cos ๐ด sin๐ต to prove the identity for sin(๐ด โ ๐ต)?
๐ฌ๐ข๐ง ๐จ โ ๐ฉ = ๐ฌ๐ข๐ง ๐จ + โ๐ฉ
= ๐ฌ๐ข๐ง ๐จ ๐๐จ๐ฌ โ๐ฉ + ๐๐จ๐ฌ ๐จ ๐ฌ๐ข๐ง โ๐ฉ= ๐ฌ๐ข๐ง๐จ ๐๐จ๐ฌ๐ฉ โ ๐๐จ๐ฌ๐จ ๐ฌ๐ข๐ง๐ฉ
What about cos(๐ด + ๐ต)? (Hint: what links ๐ ๐๐ and ๐๐๐ ?)
๐๐จ๐ฌ ๐จ + ๐ฉ = ๐ฌ๐ข๐ง๐
๐โ ๐จ + ๐ฉ = ๐ฌ๐ข๐ง
๐
๐โ ๐จ + โ๐ฉ
= ๐ฌ๐ข๐ง๐
๐โ ๐จ ๐๐จ๐ฌ โ๐ฉ + ๐๐จ๐ฌ
๐
๐โ ๐จ ๐ฌ๐ข๐ง โ๐ฉ
= ๐๐จ๐ฌ๐จ ๐๐จ๐ฌ๐ฉ โ ๐ฌ๐ข๐ง๐จ ๐ฌ๐ข๐ง๐ฉ
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?
?
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Proof of other identities
Edexcel C3 Jan 2012 Q8
The important thing is that you explicit show the division of each term by ๐๐๐ ๐ด cos๐ต.
?
Examples
Given that 2 sin(๐ฅ + ๐ฆ) = 3 cos ๐ฅ โ ๐ฆ express tan ๐ฅ in terms of tan ๐ฆ.
Using your formulae:๐ ๐ฌ๐ข๐ง๐ ๐๐จ๐ฌ ๐ + ๐๐๐จ๐ฌ ๐ ๐ฌ๐ข๐ง๐ = ๐๐๐จ๐ฌ ๐ ๐๐จ๐ฌ ๐ + ๐ ๐ฌ๐ข๐ง๐ ๐ฌ๐ข๐ง๐
We need to get ๐ญ๐๐ง ๐ and ๐ญ๐๐ง๐ in there. Dividing by ๐๐จ๐ฌ ๐ ๐๐จ๐ฌ ๐ would seem like a sensible step:
๐ ๐ญ๐๐ง ๐ + ๐ ๐ญ๐๐ง ๐ = ๐ + ๐ ๐ญ๐๐ง ๐ ๐ญ๐๐ง๐Rearranging:
๐ญ๐๐ง ๐ =๐ โ ๐ ๐ญ๐๐ง๐
๐ โ ๐ ๐ญ๐๐ง๐
Q
?
Exercises 7A
Pearson Pure Mathematics Year 2/ASPage 144
(Classes in a rush may wish to skip this exercise)
Uses of Addition Formulae
[Textbook] Using a suitable angle formulae, show that sin 15ยฐ =6โ 2
4.Q
๐ฌ๐ข๐ง๐๐ = ๐ฌ๐ข๐ง ๐๐ โ ๐๐ = ๐ฌ๐ข๐ง๐๐ ๐๐จ๐ฌ๐๐ โ ๐๐จ๐ฌ ๐๐ ๐ฌ๐ข๐ง๐๐
=๐
๐ร
๐
๐โ
๐
๐ร๐
๐
=๐ โ ๐
๐ ๐
=๐ โ ๐
๐
[Textbook] Given that sin ๐ด = โ3
5and 180ยฐ < ๐ด < 270ยฐ, and that cos๐ต = โ
12
13,
and B is obtuse, find the value of: (a) cos ๐ด โ ๐ต (b) tan ๐ด + ๐ต
๐๐๐ ๐จ โ ๐ฉ โก ๐๐๐ ๐จ ๐๐๐ ๐ฉ + ๐๐๐ ๐จ ๐๐๐ ๐ฉ
๐๐๐ ๐จ โก ๐ โ ๐๐๐๐๐จ = ยฑ๐
๐. But we know from the graph of ๐๐๐ that it is negative
between ๐๐๐ยฐ < ๐จ < ๐๐๐ยฐ, so ๐๐๐ ๐จ = โ๐
๐.
Similarly ๐๐๐ ๐ฉ = ยฑ๐
๐๐, and similarly considering the graph for ๐๐๐, ๐๐๐ ๐ฉ = +
๐
๐๐.
โด ๐๐๐ ๐จ โ ๐ฉ โก โ๐
๐ร โ
๐๐
๐๐+ โ
๐
๐ร
๐
๐๐=๐๐
๐๐
Continuedโฆ
Q
?
?
Fro Tip: You can get ๐๐๐ in terms of ๐ ๐๐ and vice versa by using a rearrangement of sin2 ๐ฅ + cos2 ๐ฅ โก 1.
So cos๐ด = 1 โ sin2 ๐ด
Continuedโฆ
Given that sin๐ด = โ3
5and 180ยฐ < ๐ด < 270ยฐ, and that cos๐ต = โ
12
13, find the
value of: (b) tan ๐ด + ๐ต
Q
๐ญ๐๐ง ๐จ + ๐ฉ =๐ญ๐๐ง ๐จ + ๐ญ๐๐ง(๐ฉ)
๐ โ ๐ญ๐๐ง๐จ ๐ญ๐๐ง๐ฉ
We earlier found ๐ฌ๐ข๐ง๐จ = โ๐
๐, ๐ฌ๐ข๐ง๐ฉ =
๐
๐๐, ๐๐จ๐ฌ๐จ = โ
๐
๐, ๐๐จ๐ฌ๐ฉ = โ
๐๐
๐๐
โด ๐ญ๐๐ง๐จ = โ๐
๐รท โ
๐
๐=
๐
๐and ๐ญ๐๐ง๐ฉ =
๐
๐๐รท โ
๐๐
๐๐= โ
๐
๐๐
By substitution:
๐ญ๐๐ง ๐จ + ๐ฉ =๐๐
๐๐
?
Test Your Understanding
Without using a calculator, determine the exact value of:a) cos 75ยฐb) tan 75ยฐ
๐๐จ๐ฌ ๐๐ยฐ = ๐๐จ๐ฌ ๐๐ยฐ + ๐๐ยฐ = ๐๐จ๐ฌ ๐๐ยฐ ๐๐จ๐ฌ ๐๐ยฐ โ ๐ฌ๐ข๐ง ๐๐ยฐ ๐ฌ๐ข๐ง ๐๐ยฐ
=๐
๐ร
๐
๐โ
๐
๐ร๐
๐
=๐ โ ๐
๐ ๐=
๐ โ ๐
๐
๐ญ๐๐ง ๐๐ยฐ =๐ฌ๐ข๐ง ๐๐ยฐ
๐๐จ๐ฌ ๐๐ยฐ๐ฌ๐ข๐ง ๐๐ยฐ = ๐ฌ๐ข๐ง ๐๐ยฐ + ๐๐ยฐ = ๐ฌ๐ข๐ง ๐๐ยฐ ๐๐จ๐ฌ ๐๐ยฐ + ๐๐จ๐ฌ ๐๐ยฐ ๐ฌ๐ข๐ง ๐๐ยฐ
=๐
๐ร
๐
๐+
๐
๐ร๐
๐=
๐ + ๐
๐
โด ๐ญ๐๐ง ๐๐ยฐ =๐ + ๐
๐รท
๐ โ ๐
๐=
๐ + ๐
๐ โ ๐= ๐ + ๐
?
?
Challenging question
Edexcel June 2013 Q3
โExpandingโ both sides:2 cos ๐ฅ cos 50 โ 2 sin ๐ฅ sin 50= sin ๐ฅ cos 40 + cos ๐ฅ sin 40
Since thing to prove only has 40 in it, use cos 50 = sin 40 and sin 50 =cos 40 .
2 cos ๐ฅ sin 40 โ 2 sin ๐ฅ cos 40= sin ๐ฅ cos 40 + cos ๐ฅ sin 40
As per usual, when we want tans, divide by cos ๐ฅ cos 40:
2 tan 40 โ 2 tan ๐ฅ = tan ๐ฅ + tan 40tan 40 = 3 tan ๐ฅ1
3tan 40 = tan ๐ฅ
a
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Exercise 7B
Pearson Pure Mathematics Year 2/ASPages 172-173
[STEP I 2010 Q3] Show thatsin ๐ฅ + ๐ฆ โ sin ๐ฅ โ ๐ฆ = 2 cos ๐ฅ sin ๐ฆ
and deduce that
sin ๐ด โ sin ๐ต = 2 cos1
2๐ด + ๐ต sin
1
2๐ด โ ๐ต
Show also that
cos๐ด โ cos๐ต = โ2 sin1
2๐ด + ๐ต sin
1
2๐ด โ ๐ต
The points ๐, ๐, ๐ and ๐ have coordinates ๐ cos ๐ , ๐ sin ๐ and ๐ cos ๐ , ๐ sin ๐ respectively,
where 0 โค ๐ < ๐ < ๐ < ๐ < 2๐, and ๐ and ๐ are positive.Given that neither of the lines ๐๐ and ๐๐ is vertical, show that these lines are parallel if and only if
๐ + ๐ โ ๐ โ ๐ = 2๐
Extension
Double Angle Formulae
sin 2๐ด โก 2 sin๐ด cos๐ดcos 2๐ด โก cos2 ๐ด โ sin2 ๐ด
โก 2 cos2 ๐ด โ 1โก 1 โ 2 sin2 ๐ด
tan 2๐ด =2 tan๐ด
1 โ tan2 ๐ด
Tip: The way I remember what way round these go is that the cos on the RHS is โattractedโ to the cos on the LHS, whereas the sin is pushed away.
These are all easily derivable by just setting ๐ด = ๐ต in the compound angle formulae. e.g.
sin 2๐ด = sin ๐ด + ๐ด= sin๐ด cos๐ด + cos๐ด sin ๐ด= 2 sin๐ด cos๐ด
!This first form is relatively rare.
Double-angle formula allow you to halve the angle within a trig function.
Examples
[Textbook] Use the double-angle formulae to write each of the following as a single trigonometric ratio.a) cos2 50ยฐ โ sin2 50ยฐ
b)2 tan
๐
6
1โtan2๐
6
c)4 sin 70ยฐ
sec 70ยฐ
This matches ๐๐จ๐ฌ ๐๐จ = ๐๐จ๐ฌ๐ ๐จ โ ๐ฌ๐ข๐ง๐๐จ:๐๐๐๐ ๐๐ยฐ โ ๐๐๐๐ ๐๐ยฐ = ๐๐จ๐ฌ ๐๐๐ยฐ
This matches ๐ญ๐๐ง ๐๐จ =๐ ๐ญ๐๐ง ๐จ
๐โ๐ญ๐๐ง๐ ๐จ
๐ ๐๐๐๐ ๐
๐ โ ๐๐๐๐๐ ๐
= ๐ญ๐๐ง๐
๐
๐ ๐ฌ๐ข๐ง๐๐ยฐ
๐ฌ๐๐ ๐๐ยฐ=
๐๐ฌ๐ข๐ง๐๐ยฐ
๐ รท ๐๐จ๐ฌ๐๐ยฐ= ๐๐ฌ๐ข๐ง๐๐ยฐ ๐๐จ๐ฌ ๐๐ยฐ
This matches ๐ฌ๐ข๐ง ๐๐จ = ๐๐ฌ๐ข๐ง๐จ ๐๐จ๐ฌ๐จ๐ ๐ฌ๐ข๐ง๐๐ยฐ ๐๐๐๐๐ยฐ = ๐ ๐ฌ๐ข๐ง๐๐๐ยฐ
sin 2๐ฅ = 2 sin ๐ฅ cos ๐ฅcos 2๐ฅ = cos2 ๐ฅ โ sin2 ๐ฅ
= 2 cos2 ๐ฅ โ 1= 1 โ 2 sin2 ๐ฅ
tan 2๐ฅ =2 tan ๐ฅ
1 โ tan2 ๐ฅ
a
b
c
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Examples
[Textbook] Given that ๐ฅ = 3 sin ๐ and ๐ฆ = 3 โ 4๐๐๐ 2๐, eliminate ๐ and express ๐ฆ in terms of ๐ฅ.
sin 2๐ฅ = 2 sin ๐ฅ cos ๐ฅcos 2๐ฅ = cos2 ๐ฅ โ sin2 ๐ฅ
= 2 cos2 ๐ฅ โ 1= 1 โ 2 sin2 ๐ฅ
tan 2๐ฅ =2 tan ๐ฅ
1 โ tan2 ๐ฅ
Fro Note: This question is an example of turning a set of parametric equations into a single Cartesian one. You will cover this in the next chapter.
๐ = ๐ โ ๐ ๐ โ ๐๐ฌ๐ข๐ง๐ ๐ฝ
= ๐๐ฌ๐ข๐ง๐ ๐ฝ โ ๐
๐ฌ๐ข๐ง๐ฝ =๐
๐
โด ๐ = ๐๐
๐
๐
โ ๐
Given that cos ๐ฅ =3
4and ๐ฅ is acute, find the exact value of
(a) sin 2๐ฅ (b) tan 2๐ฅ
Since ๐๐๐๐๐ + ๐๐จ๐ฌ๐๐ = ๐, ๐๐๐ ๐ = ยฑ ๐ โ ๐๐๐๐๐
๐๐๐ ๐ = ๐ โ๐
๐
๐=
๐
๐(and is positive given ๐ < ๐ฑ < ๐๐ยฐ)
โด ๐๐๐ ๐๐ = ๐๐๐๐ ๐ ๐๐๐ ๐ = ๐ ร๐
๐ร๐
๐=๐ ๐
๐
๐๐๐ ๐ =๐๐๐ ๐
๐๐๐=๐ ๐/๐
๐/๐=
๐
๐โด ๐๐๐๐๐ =
๐ ร๐๐
๐ โ๐๐
๐ = ๐ ๐
?
?
Solving Trigonometric Equations
This is effectively the same type of question you encountered in Chapter 6 and in Year 1, except you may need to use either the addition formulae or double angle formulae.
[Textbook] Solve 3 cos 2๐ฅ โ cos ๐ฅ + 2 = 0 for 0 โค ๐ฅ โค 360ยฐ.
3 2 cos2 ๐ฅ โ 1 โ cos ๐ฅ + 2 = 06 cos2 ๐ฅ โ 3 โ cos ๐ฅ + 2 = 06 cos2 ๐ฅ โ cos ๐ฅ โ 1 = 03 cos ๐ฅ + 1 2 cos ๐ฅ โ 1 = 0
cos ๐ฅ = โ1
3๐๐ cos ๐ฅ =
1
2
๐ฅ = 109.5ยฐ, 250.5ยฐ ๐ฅ = 60ยฐ, 300ยฐ
Recall that we have a choice of double angle identities for cos 2๐ฅ. This is clearly the most sensible choice as we end up with an equation just in terms of ๐๐๐ .?
Further Examples
[Textbook] By noting that 3๐ด = 2๐ด + ๐ด, :a) Show that sin 3๐ด = 3 sin๐ด โ 4 sin3๐ด.b) Hence or otherwise, solve, for 0 < ๐ < 2๐,
the equation 16 sin3 ๐ โ 12 sin๐ โ 2 3 = 0
๐ฌ๐ข๐ง ๐๐จ + ๐จ= ๐ฌ๐ข๐ง ๐๐จ ๐๐จ๐ฌ๐จ + ๐๐จ๐ฌ ๐๐จ ๐ฌ๐ข๐ง๐จ
= ๐ ๐ฌ๐ข๐ง๐จ ๐๐จ๐ฌ ๐จ ๐๐จ๐ฌ๐จ + ๐ โ ๐ ๐ฌ๐ข๐ง๐ ๐จ ๐ฌ๐ข๐ง๐จ
= ๐ ๐ฌ๐ข๐ง๐จ ๐ โ ๐ฌ๐ข๐ง๐ ๐จ + ๐ฌ๐ข๐ง ๐จ โ ๐๐ฌ๐ข๐ง๐ ๐จ
= ๐ ๐ฌ๐ข๐ง๐จ โ ๐ ๐ฌ๐ข๐ง๐ ๐จ + ๐ฌ๐ข๐ง๐จ โ ๐ ๐ฌ๐ข๐ง๐ ๐จ= ๐ ๐ฌ๐ข๐ง๐จ โ ๐ ๐ฌ๐ข๐ง๐ ๐จ
Note that ๐๐๐๐๐๐ ๐ฝ โ ๐๐ ๐๐๐๐ฝ
= โ๐ ๐๐ฌ๐ข๐ง ๐จ โ ๐ ๐ฌ๐ข๐ง๐ ๐จ = โ๐ ๐ฌ๐ข๐ง ๐๐ฝ.
โด โ๐ ๐ฌ๐ข๐ง ๐๐ฝ = ๐ ๐
๐ฌ๐ข๐ง ๐๐ฝ = โ๐
๐
๐๐ฝ =๐๐
๐,๐๐
๐,๐๐๐
๐,๐๐๐
๐,๐๐๐
๐,๐๐๐
๐
๐ฝ =๐๐
๐,๐๐
๐,๐๐๐
๐,๐๐๐
๐,๐๐๐
๐,๐๐๐
๐
Fro Exam Note: A question pretty much just like this came up in an exam once.
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[Textbook] Solve 4 cos ๐ โ 30ยฐ = 8 2sin ๐ in the range 0 โค ๐ < 360ยฐ.
Your instinct should be to use the addition formulae first to break up the ๐๐๐๐ ๐ฝ โ ๐๐ยฐ .
๐ ๐๐จ๐ฌ๐ฝ ๐๐จ๐ฌ๐๐ยฐ + ๐ ๐ฌ๐ข๐ง๐ฝ ๐ฌ๐ข๐ง๐๐ยฐ = ๐ ๐ ๐ฌ๐ข๐ง๐ฝ
๐ ๐ ๐๐จ๐ฌ๐ฝ + ๐๐ฌ๐ข๐ง๐ฝ = ๐ ๐๐ฌ๐ข๐ง๐ฝ
๐ ๐๐๐จ๐ฌ๐ฝ = ๐ ๐๐ฌ๐ข๐ง๐ฝ โ ๐ ๐ฌ๐ข๐ง๐ฝ
๐ ๐ ๐๐จ๐ฌ๐ฝ = ๐ ๐ โ ๐ ๐ฌ๐ข๐ง๐ฝ
๐ ๐
๐ ๐ โ ๐=๐ฌ๐ข๐ง๐ฝ
๐๐จ๐ฌ๐ฝ= ๐ญ๐๐ง๐ฝ
๐ฝ = ๐๐. ๐ยฐ, ๐๐๐. ๐ยฐ
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Test Your Understanding
Edexcel C3 Jan 2013 Q6
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If you finish that quicklyโฆ
[Textbook] Solve 2 tan2๐ฆ tan๐ฆ = 3 for 0 โค ๐ฆ < 2๐, giving your answer to 2dp.
๐ ๐ญ๐๐ง ๐๐ ๐ญ๐๐ง ๐ = ๐
๐๐ ๐ญ๐๐ง ๐
๐ โ ๐ญ๐๐ง๐ ๐๐ญ๐๐ง ๐ = ๐
โฆ๐ ๐ญ๐๐ง๐ ๐ = ๐
๐ญ๐๐ง ๐ = ยฑ๐
๐
๐ = ๐. ๐๐, ๐. ๐๐, ๐. ๐๐, ๐. ๐๐
Forgetting the ยฑ is an incredibly common source of lost marks.
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Exercise 7D
Pearson Pure Mathematics Year 2/ASPages 180-181
Extension
[AEA 2013 Q2]
1
2 [AEA 2009 Q3]
(Solution on next slide)
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๐ sin ๐ + ๐ cos ๐
Hereโs a sketch of ๐ฆ = 3 sin ๐ฅ + 4 cos ๐ฅ.What do you notice?
Itโs a sin graph that seems to be translated on the ๐-axis and stretched on the ๐ axis. This suggests we can represent it as ๐ = ๐น ๐ฌ๐ข๐ง(๐ + ๐ถ), where ๐ถ is the horizontal translation and ๐น the stretch on the ๐-axis.
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๐ sin ๐ + ๐ sin ๐
Put 3 sin ๐ฅ + 4 cos ๐ฅ in the form ๐ sin ๐ฅ + ๐ผ giving ๐ผ in degrees to 1dp.Q
Fro Tip: I recommend you follow this procedure every time โ Iโve tutored students whoโve been taught a โshortcutโ (usually skipping Step 1), and they more often than not make a mistake.
STEP 1: Expanding:๐ sin ๐ฅ + ๐ผ = ๐ sin ๐ฅ cos๐ผ + ๐ cos ๐ฅ sin ๐ผ
STEP 2: Comparing coefficients:๐ cos๐ผ = 3 ๐ sin ๐ผ = 4
STEP 3: Using the fact that ๐ 2 sin2 ๐ผ + ๐ 2 cos2 ๐ผ = ๐ 2:
๐ = 32 + 42 = 5
STEP 4: Using the fact that ๐ sin ๐ผ
๐ cos ๐ผ= tan๐ผ:
tan๐ผ =4
3๐ผ = 53.1ยฐ
STEP 5: Put values back into original expression.3 sin ๐ฅ + 4 cos ๐ฅ โก 5 sin ๐ฅ + 53.1ยฐ
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If ๐ cos ๐ผ = 3 and ๐ sin๐ผ = 4then ๐ 2 cos2 ๐ผ = 32 and ๐ 2 sin2 ๐ผ = 42.๐ 2 sin2 ๐ผ + ๐ 2 cos2 ๐ผ = 32 + 42
๐ 2 sin2๐ผ + cos2 ๐ผ = 32 + 42
๐ 2 = 32 + 42
๐ = 32 + 42
(You can write just the last line in exams)
Test Your Understanding
Put sin๐ฅ + cos๐ฅ in the form ๐ sin ๐ฅ + ๐ผ giving ๐ผ in terms of ๐.Q
๐ sin ๐ฅ + ๐ผ โก ๐ sin ๐ฅ cos ๐ผ + ๐ cos ๐ฅ sin ๐ผ๐ cos๐ผ = 1 ๐ sin๐ผ = 1
๐ = 12 + 12 = 2tan๐ผ = 1
๐ผ =๐
4
sin ๐ฅ + cos ๐ฅ โก 2 sin ๐ฅ +๐
4
Put sin๐ฅ โ 3 cos ๐ฅ in the form ๐ sin ๐ฅ โ ๐ผ giving ๐ผ in terms of ๐.Q
๐ sin ๐ฅ + ๐ผ โก ๐ sin ๐ฅ cos ๐ผ โ ๐ cos ๐ฅ sin ๐ผ
๐ cos๐ผ = 1 ๐ sin ๐ผ = 3๐ = 2
tan๐ผ = 3 ๐ ๐ ๐ผ =๐
3
sin ๐ฅ + cos ๐ฅ โก 2 sin ๐ฅ โ๐
3
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Further Examples
Put 2 cos๐ + 5 sin ๐ in the form ๐ cos ๐ โ ๐ผ where 0 < ๐ผ < 90ยฐHence solve, for 0 < ๐ < 360, the equation 2 cos๐ + 5 sin ๐ = 3
Q
2 cos ๐ + 5 sin ๐ โก 29 cos ๐ โ 68.2ยฐTherefore:
29 cos ๐ โ 68.2ยฐ = 3
cos ๐ โ 68.2ยฐ =3
29๐ โ 68.2ยฐ = โ56.1โฆ ยฐ, 56.1โฆ ยฐ๐ = 12.1ยฐ, 124.3ยฐ
(Without using calculus), find the maximum value of 12 cos๐ + 5 sin๐, and give the smallest positive value of ๐ at which it arises.
Q
Use either ๐ sin(๐ + ๐ผ) or ๐ cos(๐ โ ๐ผ) before that way the + sign in the middle matches up.
โก 13 cos ๐ โ 22.6ยฐ
๐๐๐ is at most 1,thus the expression has value at most 13.This occurs when ๐ โ 22.6 = 0 (as cos 0 = 1) thus ๐ = 22.6
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Fro Tip: This is an exam favourite!
Quickfire Maxima
What is the maximum value of the expression and determine the smallest positive value of ๐ (in degrees) at which it occurs.
Expression Maximum (Smallest) ๐ฝ at max
20 sin ๐ 20 90ยฐ
5 โ 10 sin ๐ 15 270ยฐ
3 cos ๐ + 20ยฐ 3 340ยฐ
2
10 + 3 sin ๐ โ 30
2
7
300ยฐ
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Proving Trigonometric Identities
Prove that tan 2๐ โก2
cot ๐โtan ๐
Prove that 1โcos 2๐
sin 2๐โก tan ๐
Just like Chapter 6 had โproveyโ and โsolveyโ questions, we also get the โproveyโ questions in Chapter 7. Just use the appropriate double angle or addition formula.
tan 2๐ โก2 tan ๐
1 โ tan2 ๐โก
2
1tan ๐
โ tan๐
โก2
cot ๐ โ tan๐
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1 โ 1 โ 2 sin2 ๐
2 sin ๐ cos ๐โก
2 sin2 ๐
2 sin ๐ cos ๐โก tan๐
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Test Your Understanding
[OCR] Prove that cot 2๐ฅ + ๐๐๐ ๐๐ 2๐ฅ โก cot ๐ฅ
๐๐จ๐ฌ ๐๐
๐ฌ๐ข๐ง๐๐+
๐
๐ฌ๐ข๐ง๐๐
โก๐๐จ๐ฌ ๐๐ + ๐
๐ฌ๐ข๐ง๐๐โก๐๐๐จ๐ฌ๐ ๐ โ ๐ + ๐
๐๐ฌ๐ข๐ง๐ ๐๐จ๐ฌ ๐
โก๐๐๐จ๐ฌ๐ ๐
๐ ๐ฌ๐ข๐ง๐ ๐๐จ๐ฌ ๐โก๐๐จ๐ฌ ๐
๐ฌ๐ข๐ง๐โก ๐๐จ๐ญ ๐
[OCR] By writing cos ๐ฅ = cos 2 ร๐ฅ
2or otherwise,
prove the identity 1โcos ๐ฅ
1+cos ๐ฅโก tan2
๐ฅ
2
๐๐จ๐ฌ ๐ โก ๐๐๐ ๐ ร๐
๐โก ๐๐๐จ๐ฌ๐
๐
๐โ ๐
โด๐ โ ๐๐๐ ๐
๐ + ๐๐๐ ๐โก๐ โ ๐๐๐จ๐ฌ๐
๐๐
โ ๐
๐ + ๐๐๐๐๐๐๐
โ ๐
โก๐ ๐ โ ๐๐จ๐ฌ๐
๐๐
๐ ๐๐จ๐ฌ๐๐๐
โก๐ฌ๐ข๐ง๐
๐๐
๐๐จ๐ฌ๐๐๐
โก ๐ญ๐๐ง๐๐
๐
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Exercise 7F
Pearson Pure Mathematics Year 2/ASPages 187-188
Extension:
[STEP I 2005 Q4]
(a) โ117
125(b) ๐ก๐๐ ๐ = 8 + 75? ?
Very Challenging Exam Example
Edexcel C3 June 2015 Q8
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We eventually want cos ๐ด โ sin๐ดso cos2 ๐ด โ sin2๐ด is best choice of double-angle formula because this can be factorise to give cos ๐ด โ sin๐ด as a factor.
This is even less obvious. Knowing that weโll have cos๐ด โ sin ๐ด cos๐ด + sin ๐ด in the denominator and that the cos๐ด + sin ๐ด will cancel, we might work backwards from the final result and multiply by cos๐ด + sin ๐ด! Working backwards from the thing weโre trying to prove is occasionally a good strategy (provided the steps are reversible!)
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Modelling
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When trigonometric equations are in the form ๐ ๐ ๐๐ ๐๐ฅ ยฑ ๐ or ๐ ๐๐๐ ๐๐ฅ ยฑ ๐ , they can be used to model various things which have an oscillating behaviour, e.g. tides, the swing of a pendulum and sound waves.
Test Your Understanding
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Tip: Reflect carefully on the substitution you use to allow (bii) to match your identity in (a). ๐ =?