Trigonometry for Scalene Triangles and Trigonometric Equations

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formula so COS A is the subject 26c Trigonometry with Bearings! 3 hey points for finding bearings. 1) The angle is always measured starting from north (measure noAn line)) 2) Bearings are aways measured dockwise 3) Bearings alterays have 3 figures (add Os) ↳ shows it's the whole bearing (not part) Examples 1) A ship A is 10 um from a lighthouse L on a bearing of 115⁰ Another shippis 14.5hm from L on a bearing of 200°. 70 14.5mm B 28hm z = 360 = 11 X US 850 10hm 284,4° (284 iF 3 sf) 32um xem là 2) An explorer walls 28hm on a bearing of 070° and then she walls 32 honch a bearing of 134º. N y 240 46 - 29.6 A x² 8 8 8. X 102 +14.5 284.978.. = 16.9 km (to 3 sf) = x2 X 2 siny 28 2x10x14.5xcos 85 28² +32²-2x28x32x COS116 2593. 56.. 50.9m (to 3sf) Sin 116 50.9 Sin (sin 116 50.9 y = 29.6 (to 3st) x28) 39 Finding the area of triangles without perpendicular neight| A C b sector 7 Examples 40 h Sin C = h b h = b sin C 8cm Example a A 10.4 cm 10cm Area of a segment = area of sector C B e 1200 O B ABC an arc 10.4 CA Area of triangle Area = Area = Area of secter = 120 360 B Area = /2x10x 8x Sin 43 = 27. 3 cm² (to 3sf) D 1/2 bh 2 absih C Area rule (area of area of triangle AREShaded a) Calculate the length of the arc ABC •C a segment C Arck length 360° 120 non-right angled trangle) x 2 TTC b) Calculate the area of the shaded segrent ABC area = TTC² x TT x 10.4² x 2TT x 10.4 360 = 21.8 cm 40 3sf.) Area of segrent = 113.26-46.83 = 1/2 x 10.42 x Sin 120 = 113.26cm² = 46.83 cm² = 66.4 cm² Frigonometry Grapns (see large on loose paper) X (C XQg sinx y lo need to /now these y = cosx y . XO 90 180 270 1 a -P y = tanx 180 90 1 O values x 0 90 180 yo ? 0 (8) sin C AB Sin C 10 C AB • never goes above I below-1 (2) so Ac BC 270 -1 4 opposite for negathe = AC same as sin, but translated 90° in the p-x direction • never goes above I below-1 Sin B AC 270 ? (00) SM30 7 360 O 360 e 45.6° Obtuse angle at C 1-360 360 0 -270 sin (Sim 30 x 10) 7 B - 360 draw these first "Sin wowe) - 189 AB Find two possible values of LACB. (2) cos wage? buchet 7 Sine Rule Ambiguity A ABC has LABC = 30°, AB = 10cm, and AC = 7cm. Shetch two triangles that meet these criteria. -90 about y axis - 270 - 180 40 -270 BC 09 symenetically -180 AC -90 90 с 180- 45.6 0 10 = ware % (stands line this acute angle at 180 repeat every 180°) 90 40 (surve 134.4° 4 should be smoother + noffe 270 180 210 'bucket C - calculater always gives This X Generally find acture langle, just Subtract from) 180° to get obtuse one 270 360 sin & COS -repeat every 360° 360 Trigonometry Equations Solve for O sin = 0.3 esin (0.3) 9 = 17.5" because of sin rule ambiguity... D O 42 cos = 1-2-222 e = cos(-1/₂) 0=120° 0 = 240° (n.) 90 180 270 360 see from symetry how 180-0 = other value -180-17.5 = 162.5° = O≤ ≤ 360° (for iGCSE) 130 e- tane = 4 6 = tan-¹ (4) Ө 75.0⁰ T 120 (30 theta often used for ununoun angles NI 1270 draw sin wave 360 between 0 and 360 270360 generals speaking, helps to arawa quich sketch 30 in from 90 = 120° 30 in from 270 = 240° OR 360 120 240° = 180 + 76 = 2560 (1200 difference) (or 270-(90-76)) Slightly different (for each-drawing. out graphs move efficient More Difficult Examples (a) 6 sine + 5 = 0 6 sine =-5 -56.4 D O Sin=-S -5 6 = @ = ³n-¹ ( - {) Sin e-56.4 236,4, 303.6 (b) Sin 20 = 0.55 20 го 180 -33.7 1100 130+ 56+ Identities 180 sin (0.55) = 33.37, 146.3,393.7, 506.3 360 360-56.4 16.7 73.2, 197,253 3x+2 360 equation → term (word) - 3x =Something that is always tre Chonater what! value of sc) 360 4337.7 540.-33.7 (Tips + remember if squared sin (sin? ☺), etc., two possible values of IVI, solve for both !! I could need to solve quadratic → x becomes Foo = 2x - 4 expression → 3x +2 (sentence) identity 3 C +2) = 3sc +6 04 204 726 * see examples in exercise book to -anc true for certain values ↑ both sides cavally 100 sine =shows identity 43 Two trig identities: (x, y) right angled (height ( 441 0 x tane sin e COSO sin 20 + cos 2 ... COS² Sin 2 = 1 21- Sine= tan cose cose = sin @ tan o = 0 = ² cos²¹ (2) 60 e = 60, 300 2 sin 2 sin G cose 2 Examples (to save equations) (a) 2sin+ 3 cos - 2 2 (201-Cos²0) + 3 cos cose = 1 2tan = 2 tane 0 1 tah @ = xy y Si COS Identity 1 2-2 cos20 + 3 cose - 2 cos ² + 3 cos @ -1=0 2cos² - 3 cose + 1 = 0 (2cos -1) (cos@ - 1) = 0 Cose or COS 0 = 1 1 Identity 2 Sin e COS? O = y cas radius () 2 tan-'( =) 26.565 =0 - cos-1(1) 360 0 00 1 allow Convesions fraue - con manipulate trig expresons + solve equations (2x²-3x+1=6) 180 360 180+26.565 = 206.565 26.6 206.6 Ⓒ=26.6 or if 040 <180 A