C3 Trigonometry ANSWERS



C3 Trigonometry ANSWERS

|4. | |[pic] |

| |[pic] | |

| | | [pic] * |M1 A1 (4) |

|2. |(a) |sin x + (3 cos x = R sin (x + () | |

| | | = R (sin x cos ( + cos x sin () |M1 |

| | |R cos ( = 1, R sin ( = (3 |A1 |

| | |Method for R or (, e.g. R = ((1 + 3) or tan ( = (3 |M1 |

| | |Both R = 2 and ( = 60 |A1 (4) |

| |(b) | sec x + (3 cosec x = 4 ( [pic] + [pic] = 4 |B1 |

| | |( sin x + (3 cos x = 4 sin x cos x |M1 |

| | | = 2 sin 2x (*) |M1 (3) |

| |(c) |Clearly producing 2 sin 2x = 2 sin (x + 60) |A1 (1) |

| |(d) |sin 2x – sin (x + 60) = 0 ( cos [pic] sin [pic] = 0 |M1 |

| | |cos [pic] = 0 ( x = 40(, 160( |M1 A1 A1 ft |

| | |sin [pic] = 0 ( x = 60( |B1 (5) |

|3. |(a) |12 cos ( ( 5 sin ( = R cos ( cos( ( R sin ( sin ( | |

| | | |R2 = 52 + 122, ( R = 13 | |

| | | | | |

| | | |tan ( = [pic], ( ( = 22.6( (awrt 22.6) |M1, A1 (4) |

| | | |(AWRT or 0.39c (AWRT 0.39c) | |

| | | | | |

| |(b) |cos (( + 22.6) = [pic] |M1 |

| | |( + 22.6 = 72.1, |M1 |

| | | ( = 49.5 (only) |A1 (3) |

| |(ii) | [pic] |M1 |

| | |i.e. 0 = 3tan2[pic] + 2tan[pic] |M1 |

| | | [pic] |M1 |

| | | [pic] | |

| | | [pic]( = 53.1 |A1 |

| | | [pic][pic]] |A1 (5) |

|4. |(i) | A correct form of cos 2x used |M1 |

| | | 1 – 2[pic] or [pic] or [pic] [pic] |A1 |

| | |[pic] ; [pic] or [pic] |M1A1 (4) |

| |(ii) |(a) [pic]or (b) [pic] |M1 |

| | |Forming single fraction (or multiplying both sides by sin2x) |M1 |

| | |Use of correct trig. formulae throughout and producing expression in terms of sin x and cos x |M1 |

| | |Completion (cso) e.g. [pic] = [pic] (*) |A1 (4) |

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