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P 1 Chapter 10 : : Trigonometric Identities & Equations [email protected] kingston. sch. uk www. drfrostmaths. com @Dr. Frost. Maths Last modified: 3 rd May 2020
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Chapter Overview 4: : Solve equations which are quadratic in sin/cos/tan.
? 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. ? ? ? ? ? ? ?
The Unit Circle and Trigonometry ? ? Angles are always measured anticlockwise. ? (Further Mathematicians will encounter the same when they get to Complex Numbers)
Mini-Exercise ? ? 1 0 0 +ve? +ve 0 1 -ve +ve ? ? Undefined (vertical lines don’t have a well-defined gradient) -ve ? ? -1 ? 0 0 -ve ? +ve 0 ? -1 +ve ? -ve Undefined -ve
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. 1 We saw this in the previous chapter when covering the ‘ambiguous case’ when using the sine rule. 2 3 4 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 C 3 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.
Test Your Understanding Without a calculator, work out the value of each below. ? ? ?
Exercise 10 A/B Pearson Pure Mathematics Year 1/AS Page 207, 209
Trigonometric Identities Returning to our point on the unit circle… ? ? 1 2 You are really uncool if you get this reference. ? Pythagoras gives you. . . ?
Application of identities #1: Proofs ? ? Fro Tip #1: Turn any tan’s into sin’s and cos’s.
More Examples Edexcel C 2 June 2012 Paper 1 Q 16 ? ? Fro Tip #2: In any addition/subtraction involving at least one fraction (with trig functions), always combine algebraically into one. ?
Test Your Understanding ? ? AQA IGCSE Further Maths Worksheet ?
Exercise 10 C Pearson Pure Mathematics Year 1/AS Page 211 -212 Extension: ?
Solving Trigonometric Equations Remember those trigonometric angle laws (on the right) earlier this chapter? They’re about to become super freakin’ useful! ? ?
Slightly Harder Ones… ? ? 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?
Test Your Understanding ? ?
Exercise 10 D Pearson Pure Mathematics Year 1/AS Page 215 -216
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.
Further Examples ? ?
Test Your Understanding Edexcel C 2 Jan 2013 Q 4 ?
Exercise 10 E Pearson Pure Mathematics Year 1/AS Page 218 -219
Quadratics in sin/cos/tan ? ? Fropinion: I’d definitely advocate Method 2 provided you feel confident with it. Method 1 feels clunky.
More Examples Missing the negative case would result in the loss of multiple marks. Beware! ? ?
Test Your Understanding Edexcel C 2 Jan 2010 Q 2 ?
Exercise 10 F Pearson Pure Mathematics Year 1/AS Page 221 -222 Extension 1 2 ? ?