WJEC Mathematics for AS Level - Pure

7 How to use this book • Step by steps are included to help you answer questions that do not guide you bit by bit towards the ϐinal answer (called unstructured questions). In the past, you would be guided to the ϐinal answer by the question being structured. For example, there may have been parts (a), (b), (c) and (d). Now you can get questions which ask you to go to the answer to part (d) on your own. You have to work out for yourself the steps (a), (b) and (c) you would need to take to arrive at the ϐinal answer. The ‘step by steps’ help teach you to look carefully at the question to analyse what steps need to be completed in order to arrive at the answer. • Active learning – are short tasks which you carry out on your own which aid understanding of a topic or help with revision. • Summaries – are provided for each topic and present the formulae and the main points in a topic. They can be used for quick reference or help with your revision. Exam Practice and Technique Helping you understand how to answer examination questions lies at the heart of this book. This means that we have included questions throughout the book that will build up your skills and knowledge until you are at a stage to answer full exam questions on your own. Examples are included; some of which are full examination style questions. These are annotated with Pointers and general advice about the knowledge, skills and techniques needed to answer them. Test yourself answers 192 Topic 5 ➊ ȋƒȌ ”‡ƒα ͳ ʹ bc Ԝ•‹Ԝ A α ͳ ʹ έͳʹέͺέ•‹ԜͳͷͲιαʹͶԜ  ʹ ȋ„Ȍ •‹‰…‘•‹‡”—Ž‡ a ʹ α b ʹ Ϊ c ʹ Ϋʹ bc Ԝ…‘•Ԝ A αͳʹ ʹ Ϊͺ ʹ ΫʹέͳʹέͺԜ…‘•ԜͳͷͲια͵͹ͶǤʹ͹͸ͻ a αͳͻǤ͵Ͷ͸ʹ a αͳͻǤ͵Ԝ ȋ–‘‘‡†‡…‹ƒŽ’Žƒ…‡Ȍ ➋ ȋƒȌ ȋͲǡͲȌǡȋɎǡͲȌǡȋʹɎǡͲȌǡȋ͵ɎǡͲȌǡȋͶɎǡͲȌ ȋ„Ȍ ȋ Ɏ ʹ ǡͳ Ȍ ǡ ȋ ͵Ɏ ʹ ǡΫͳ Ȍ ǡ ȋ ͷɎ ʹ ǡͳ Ȍ ǡ ȋ ͹Ɏ ʹ ǡΫͳ Ȍ ➌ Ͷ… ȍ͵ ඥ ʹΫͳȎ… ͵Ͳι •‹‰–Š‡•‹‡”—Ž‡ •‹Ԝ C c α •‹Ԝ A a •‹Ԝ ෝ Ͷ α •‹Ԝ͵Ͳι ͵ √ ʹΫͳ •‹Ԝ ෝ α ͶԜ•‹Ԝ͵Ͳι ͵ √ ʹΫͳ •‹Ԝ ෝ α ʹ ͵ √ ʹΫͳ α ʹȍ͵ √ ʹΪͳȎ ȍ͵ √ ʹΫͳȎȍ͵ √ ʹΪͳȎ •‹Ԝ ෝ α ͸ √ ʹΪʹ ͳͺΫͳ α ʹΪ͸ √ ʹ ͳ͹ ➍ ͵Ԝ•‹ ʹ Ԝ Ʌ αͷΫͷԜ…‘•Ԝ Ʌ ͵ȍͳΫ…‘• ʹ Ԝ Ʌ ԜȎαͷΫͷԜ…‘•Ԝ Ʌ ͵Ϋ͵Ԝ…‘• ʹ Ԝ Ʌ αͷΫͷԜ…‘•Ԝ Ʌ ͵Ԝ…‘• ʹ Ԝ Ʌ ΫͷԜ…‘•Ԝ Ʌ ΪʹαͲ ȍ…‘•Ԝ Ʌ ΫͳȎȍ͵Ԝ…‘•Ԝ Ʌ ΫʹȎαͲ …‘•Ԝ Ʌ αͳԙ‘”ԙ…‘•Ԝ Ʌ α ʹ ͵ Ʌ α…‘• Ϋͳ ԜȋͳȌαͲιǡ͵͸ͲιǡԘ‘”Ԙ Ʌ α…‘• Ϋͳ Ԝ ȋ ʹ ͵ Ȍ αͶͺǤʹιǡȋ͵͸ͲΫͶͺǤʹȌι ‡…‡ Ʌ αͲιǡͶͺǤʹιǡ͵ͳͳǤͺι‘”͵͸Ͳι ➎ ȋƒȌ •‹‰…‘•‹‡”—Ž‡ǡ™‡Šƒ˜‡ǣ ȋ x ΪʹȌ ʹ α x ʹ Ϊȋ x ΫʹȌ ʹ Ϋʹέ x έ ȋ x ΫʹȌԜ…‘•Ԝ ෝ x ʹ ΪͶ x ΪͶα x ʹ Ϊ x ʹ ΫͶ x ΪͶΫʹ x ȋ x ΫʹȌԜ…‘•Ԝ ෝ Ϋ x ʹ Ϊͺ x αΫʹ x ȋ x ΫʹȌԜ…‘•Ԝ ෝ …‘•Ԝ ෝ α x ȍͺΫ x Ȏ Ϋʹšȋ x ΫʹȌ α ΫȍͺΫ x Ȏ ʹȋ x ΫʹȌ α x Ϋͺ ʹ x ΫͶ ‡‡„‡”‘––‘”‘—† ‘ˆˆƒ•™‡”•–‘–Š‡”‡“—‹”‡† —„‡”‘ˆ†‡…‹ƒŽ’Žƒ…‡• —–‹Ž–Š‡ϐ‹ƒŽƒ•™‡”Ǥ ‘†‹ƒ‰”ƒ‹•‰‹˜‡‹–Š‡ “—‡•–‹‘ǡ•‘†”ƒ™›‘—”‘™ ƒ†ƒ††–Š‡‹ˆ‘”ƒ–‹‘Ǥ ‘— …ƒ‘™•‡‡–Š‡’ƒ‹”‹‰‘ˆ –Š‡•‹†‡•ƒ†ƒ‰Ž‡•™Š‹…Š ’‘‹–•–‘–Š‡‡‡†–‘—•‡–Š‡ •‹‡”—Ž‡Ǥ ‡‘˜‡–Š‡•—”†ˆ”‘ –Š‡†‡‘‹ƒ–‘”„› —Ž–‹’Ž›‹‰–Š‡—‡”ƒ–‘” ƒ††‡‘‹ƒ–‘”„› –Š‡…‘Œ—‰ƒ–‡‘ˆ–Š‡ †‡‘‹ƒ–‘”ǡ‹Ǥ‡Ǥ ȋ ͵ √ ʹ᩿Ϊ᩿ͳ Ȍ Ǥ •‡–Š‡–”‹‰‹†‡–‹–› …‘• ʹ᩿ Ʌ ᩿Ϊ᩿•‹ ʹ᩿ Ʌ ᩿αͳ–‘…”‡ƒ–‡ƒ “—ƒ†”ƒ–‹…‡“—ƒ–‹‘‹…‘• ᩿ Ʌ ™Š‹…Š…ƒ–Š‡„‡•‘Ž˜‡†Ǥ _WJECMaths_ASPure.indb 192 5 Trigonometry 112 Test yourself ➊ Š‡†‹ƒ‰”ƒ•Š‘™•–Š‡–”‹ƒ‰Ž‡ ™‹–Š αͺԜ…ǡ αͳʹԜ…ƒ† ෝ ᩿α᩿ͳͷͲιǤ ͳʹ… ͺ… ȋƒȌ ‹†–Š‡ƒ”‡ƒ‘ˆ–”‹ƒ‰Ž‡ Ǥ ȋ„Ȍ ‹†–Š‡Ž‡‰–Š‘ˆ …‘””‡…––‘‘‡†‡…‹ƒŽ’Žƒ…‡Ǥ ➋ Š‡‰”ƒ’Š•Š‘™•–Š‡…—”˜‡ y α•‹Ԝ x ‹–Š‡‹–‡”˜ƒŽͲζ x ζͶɎ y x Ͳ y α•‹ x ȋƒȌ ”‹–‡†‘™–Š‡…‘‘”†‹ƒ–‡•‘ˆƒŽŽ–Š‡’‘‹–•‘ˆ‹–‡”•‡…–‹‘™‹–Š–Š‡ x Ǧƒš‹•Ǥ ȋ„Ȍ ”‹–‡†‘™–Š‡…‘‘”†‹ƒ–‡•‘ˆƒŽŽ•–ƒ–‹‘ƒ”›’‘‹–•ˆ‘”–Š‹•‰”ƒ’ŠǤ ➌ Š‡–”‹ƒ‰Ž‡ ‹••—…Š–Šƒ– αͶԜ…ǡ αȋ͵ √ ʹΫͳȌԜ…ƒ† ෝ ᩿α᩿͵ͲιǤ ‹†ƒ‡š’”‡••‹‘ˆ‘”•‹Ԝ ෝ ‹–Š‡ˆ‘” ʹΪ m √ ʹ n Ԝǡ™Š‡”‡ m ƒ† n ƒ”‡ ‹–‡‰‡”•™Š‘•‡˜ƒŽ—‡•ƒ”‡–‘„‡ˆ‘—†Ǥ ➍ ‹†ƒŽŽ–Š‡˜ƒŽ—‡•‘ˆ Ʌ ‹–Š‡”ƒ‰‡Ͳιζ Ʌ ζ͵͸Ͳ镃–‹•ˆ›‹‰ ͵Ԝ•‹ ʹ Ԝ Ʌ αͷΫͷԜ…‘•Ԝ Ʌ ➎ Š‡†‹ƒ‰”ƒ„‡Ž‘™•Š‘™•ƒ•‡–…Š‘ˆ–Š‡–”‹ƒ‰Ž‡ ™‹–Š α x Ԝ…ǡ αȋ x ΫʹȌԜ…ƒ† αȋ x ΪʹȌԜ…Ǥ x … ȋ x ΪʹȌ… ȋ x ΫʹȌ… ȋƒȌ Š‘™–Šƒ–…‘•Ԝ ෝ α x Ϋͺ ʹ x ΫͶ ԜǤ ȋ„Ȍ ‹˜‡–Šƒ– ෝ αͳʹͲ ι ǡϐ‹†–Š‡•‹œ‡‘ˆƒ‰Ž‡ ෝ Ǥ ➏ ȋƒȌ ‹†ƒŽŽ˜ƒŽ—‡•‘ˆ Ʌ ‹–Š‡”ƒ‰‡Ͳιζ x ζ͵͸Ͳ镃–‹•ˆ›‹‰ ͶԜ…‘• ʹ Ԝ Ʌ ΪͳαͶԜ•‹ ʹ Ʌ ΫʹԜ…‘•Ԝ Ʌ 13/10/2017 10:27