P2 Chapter 6 :: �Trigonometric Identities & Equations
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Chapter Overview
4:: Solve equations which are quadratic in sin/cos/tan.
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Although you will always have a calculator, you need to know how to derive these.
All you need to remember:
🖉 Draw half a unit square and half an equilateral triangle of side 2.
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The Unit Circle and Trigonometry
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Angles are always measured anticlockwise.�(Further Mathematicians will encounter the same when they get to Complex Numbers)
Mini-Exercise
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(vertical lines don’t have a well-defined gradient)
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The Unit Circle and Trigonometry
Note: The textbook uses something called ‘CAST diagrams’. I will not be using them in these slides, but you may wish to look at these technique as an alternative approach to various problems in the chapter.
A Few Trigonometric Angle Laws
The following are all easily derivable using a quick sketch of a trigonometric graph, and are merely a convenience so you don’t always have to draw out a graph every time.
You are highly encouraged to memorise these so that you can do exam questions faster.
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We saw this in the previous chapter when covering the ‘ambiguous case’ when using the sine rule.
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Remember from the previous chapter that “cosine” by definition is the sine of the “complementary” angle.
This was/is never covered in the textbook but caught everyone by surprise when it came up in a C3 exam.
Examples
Without a calculator, work out the value of each below.
We have to resort to a sketch for this one.
Again, let’s just use a graph.
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Test Your Understanding
Without a calculator, work out the value of each below.
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Exercise 6A/B
Pearson Pure Mathematics 1
Page 118,120
Trigonometric Identities
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Pythagoras gives you...
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Returning to our point on the unit circle…
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You are really uncool if you get this reference.
Application of identities #1: Proofs
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Fro Tip #1: Turn any tan’s into sin’s and cos’s.
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More Examples
Edexcel C2 June 2012 Paper 1 Q16
Fro Tip #2: In any addition/subtraction involving at least one fraction (with trig functions), always combine algebraically into one.
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Test Your Understanding
AQA IGCSE Further Maths Worksheet
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Exercise 6C
Pearson Pure Mathematics 1
Page 122-123
Extension:
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Solving Trigonometric Equations
Remember those trigonometric angle laws (on the right) earlier this chapter? They’re about to become super freakin’ useful!
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Slightly Harder Ones…
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Hint: The problem here is that we have two different trig functions. Is there anything we can divide both sides by so we only have one trig function?
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Test Your Understanding
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Exercise 6D
Pearson Pure Mathematics 1
Page 126-127
Harder Equations
STEP 2: Immediately after applying an inverse trig function (and BEFORE dividing by 3!), find all solutions up to the end of the interval.
STEP 3: Then do final manipulation to each value.
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Further Examples
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Test Your Understanding
Edexcel C2 Jan 2013 Q4
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Exercise 6E
Pearson Pure Mathematics Year 1/AS
Page 129-130
Quadratics in sin/cos/tan
Fropinion: I’d definitely advocate Method 2 provided you feel confident with it. Method 1 feels clunky.
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More Examples
Missing the negative case would result in the loss of multiple marks. Beware!
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Test Your Understanding
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Edexcel C2 Jan 2010 Q2
Exercise 6F
Pearson Pure Mathematics 1
Page 132-133
Extension
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