Grade 11 TRIGONOMETRY here Tara Conradie 23 July 2022 TRIGONOMETRY NB - make sure your calculator is on ‘DEG’ 1. DEFINITIONS, IDENTITIES & RULES: 1.1 Definitions: 𝑌 For any size 𝜃 : 𝑦 ⚫ sin 𝜃 = ⚫ cos 𝜃 = 𝑟 ⚫ tan 𝜃 = 𝑥 𝑟 𝑥 𝑦 (𝑥 ; 𝑦) = ⚫ 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑟 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝜃 𝑋 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 1.2 Identities: sin 𝜃 1 tan 𝜃 = cos 𝜃 ⚫ ⚫ sin2 𝜃 + cos2 𝜃 = 1 ⚫ sin2 𝜃 = 1 − cos2 𝜃 ⚫ cos2 𝜃 = 1 − sin2 𝜃 tan 𝜃 = cos 𝜃 ⚫ sin 𝜃 1.3 180° Rule: 90° 1) Determine the quadrant 2) Use CAST diagram to determine the sign 3) Keep the same ratio (sin, cos or tan), but possibly change the sign. e.g. sin(−180° + 𝜃 ) = − sin 𝜃 because (−180° + 𝜃) is in the 3rd quadrant, where sine is negative 180° − 𝜃 180° 𝜃 𝑆 𝐴 𝑇 𝐶 180° + 𝜃 0° 360° 360° − 𝜃 270° 1.4 90° Rule: ⚫ ⚫ sin(90° − 𝜃 ) = cos 𝜃 cos(90° − 𝜃 ) = sin 𝜃 ⚫ ⚫ + sin(90° + 𝜃 ) = cos 𝜃 cos(90° + 𝜃 ) = −sin 𝜃 − 2. SPECIAL ANGLES: 2.1 Special triangles: Know the 45°/30°/60° ratios without a calculator. Use calculator for 0°/90°/180°/270°/360° ⚫ sin 45° = 1 √2 ⚫ cos 45° = 1 √2 √2 1 45° 1 30° ⚫ tan 45° = 1 2 ⚫ sin 60° = √3 2 ⚫ cos 60 ° = 1 2 ⚫ tan 60° = √3 ⚫ sin 30° = 1 2 √3 ⚫ cos 30 ° = √3 2 ⚫ tan 30° = 1 √3 60° 1 2.2 Combined with 180° and 90° rules: e.g. cos 150° = − cos 30° ALWAYS first reduce to an acute angle. = − √3 2 3. USING SKETCHES: Y 3.1 If 5 cos 𝜃 = −4 , use a sketch (and not a calculator) to find sin 𝜃 if 180° ≤ 𝜃 ≤ 360° ⚫ ⚫ ⚫ 𝜃 X Must be quadrant 3 where cos is negative. Use Pythagoras to calculate 𝑦. Watch out for the sign. 3.2 If tan 20° = 𝑝 3 ⚫ ⚫ 3 𝑦 = −3 , therefore sin 𝜃 = − 5 , write sin2 20° in terms √𝑝 2 + 9 of 𝑝 . ⚫ 5 (−4 ; 𝑦) ⚫ 𝑝 20° Draw a right-angled triangle and fill in 𝑝 and 3 on the opposite and adjacent sides. 3 rd ⚫ Use Pythagoras to calculate the 3 side. 2 sin 20° = ( 𝑝 √𝑝2 +9 2 ) = 𝑝2 𝑝2 +9 4. TRIG EQUATIONS: 4.1 ⚫ Equations must be manipulated to give: ⚫ Watch out for −1 ≤ the number ≤ 1 for sin / cos 𝜃 ⚫ First find the help-angle: ⚫ Then use CAST-diagram to determine quadrants where positive or negative. ⚫ sin / cos / tan 𝜃 = a number ⚫ SHIFT-button (don’t push minus) sin / cos : + 360°𝑛 ; 𝑛 ∈ ℤ tan : + 180°𝑛 ; 𝑛 ∈ ℤ Find all angles in interval by substituting 𝑛 = 1 , −1 , 2 , etc 4.2 ⚫ General solution: ⚫ ⚫ ⚫ Interval given: ⚫ 4.3 ⚫ sin 𝛼 = 3 cos 𝛼 ⚫ Divide by cos 𝛼 to get tan 𝛼 ⚫ sin 𝛼 = cos 𝛽 ⚫ Use the 90° rule to change sin ↔ cos 4.4 Factorising: eg. eg. eg. eg. sin 𝑥 + 3 sin 𝑥 cos 𝑥 = 0 sin2 𝑥 − 9cos2 𝑥 = 0 sin2 𝑥 − 2 sin 𝑥 cos 𝑥 + cos2 𝑥 = 0 6 cos 𝑥 − 3 cos2 𝑥 + 2 sin 𝑥 cos 𝑥 − 4 sin 𝑥 = 0 ⚫ ⚫ ⚫ ⚫ ⚫ Common factor Difference of squares Trinomial Grouping Sum & Difference of cubes TRIGONOMETRIE NB – maak seker jou sakrekenaar is op ‘DEG’ 1. DEFINISIES, IDENTITEITE & REËLS: 1.1 Definisies: 𝑌 Vir enige grootte 𝜃 : 𝑦 ⚫ sin 𝜃 = ⚫ cos 𝜃 = 𝑟 ⚫ tan 𝜃 = 𝑟 𝑥 𝑦 𝑥 = (𝑥 ; 𝑦) ⚫ 𝑜𝑜𝑟𝑠𝑡𝑎𝑎𝑛𝑑𝑒 𝑟 𝑠𝑘𝑢𝑖𝑛𝑠𝑠𝑦 𝜃 𝑋 𝑎𝑎𝑛𝑔𝑟𝑒𝑛𝑠𝑒𝑛𝑑 = 𝑠𝑘𝑢𝑖𝑛𝑠𝑠𝑦 𝑜𝑜𝑟𝑠𝑡𝑎𝑎𝑛𝑑𝑒 = 𝑎𝑎𝑛𝑔𝑟𝑒𝑛𝑠𝑒𝑛𝑑 1.2 Identiteite: sin 𝜃 1 cos 𝜃 ⚫ tan 𝜃 = cos 𝜃 ⚫ ⚫ sin2 𝜃 + cos2 𝜃 = 1 ⚫ sin2 𝜃 = 1 − cos2 𝜃 ⚫ cos2 𝜃 = 1 − sin2 𝜃 tan 𝜃 = sin 𝜃 1.3 180° reël: 90° 1) Bepaal die kwadrant 2) Gebruik CAST-diagram om teken te bepaal 3) Behou dieselfde verhouding (sin, cos of tan), maar verander dalk die teken. bv. sin(−180° + 𝜃 ) = − sin 𝜃 want (−180° + 𝜃) is in die 3de kwadrant, waar sin negatief is 180° − 𝜃 180° 𝜃 𝑆 𝐴 𝑇 𝐶 180° + 𝜃 0° 360° 360° − 𝜃 270° 1.4 90° reël: ⚫ ⚫ sin(90° − 𝜃 ) = cos 𝜃 cos(90° − 𝜃 ) = sin 𝜃 ⚫ ⚫ + sin(90° + 𝜃 ) = cos 𝜃 cos(90° + 𝜃 ) = −sin 𝜃 − 2. SPESIALE HOEKE: 2.1 Spesiale driehoeke: Ken die 45°/30°/60° verhoudings sonder ‘n sakrekenaar. Gebruik ‘n sakrekenaar vir 0°/90°/180°/270°/360° ⚫ sin 45° = 1 √2 ⚫ cos 45° = 1 √2 √2 1 45° 1 30° 2 ⚫ tan 45° = 1 √3 ⚫ sin 60° = √3 2 ⚫ cos 60 ° = 1 2 ⚫ tan 60° = √3 ⚫ sin 30° = 1 2 ⚫ cos 30 ° = √3 2 ⚫ tan 30° = 1 √3 60° 1 2.2 Met 180° en 90° reëls gekombineer: bv. cos 150° = − cos 30° Maak ALTYD eerste ‘n skerphoek. = − √3 2 3. GEBRUIK VAN SKETSE: Y 3.1 As 5 cos 𝜃 = −4 , gebruik ‘n skets (en nie ‘n sakrekenaar) om sin 𝜃 te vind as 180° ≤ 𝜃 ≤ 360° ⚫ ⚫ ⚫ 𝜃 X Moet kwadrant 3 wees waar cos negatief is. Gebruik Pythagoras om 𝑦 te bereken. Oppas vir die teken. (−4 ; 𝑦) ⚫ ⚫ 5 3 𝑦 = −3 , dus is sin 𝜃 = − 5 𝑝 3.2 As tan 20° = 3 , skryf sin2 20° in terme van 𝑝 . ⚫ ⚫ √𝑝 2 + 9 Teken ‘n reghoekige driehoek en vul 𝑝 en 3 op die teenoorstaande en aangrensende sye in. 𝑝 20° 3 Gebruik Pythagoras om die 3de sy te vind. ⚫ 2 sin 20° = ( 𝑝 √𝑝2 +9 2 ) = 𝑝2 𝑝2 +9 4. TRIGONOMETRIESE VERGELYKINGS: 4.1 ⚫ Vergelykings moet herrangskik word as: ⚫ Oppas vir −1 ≤ die getal ≤ 1 vir sin / cos 𝜃 ⚫ Vind eerstens die hulphoek: ⚫ Gebruik dan die CAST-diagram om die positiewe/negatiewe kwadrante te bepaal. ⚫ sin / cos / tan 𝜃 = ‘n getal ⚫ SHIFT-knoppie (moenie minus druk nie) 4.2 ⚫ Algemene oplossing: ⚫ ⚫ sin / cos ∶ + 360°𝑛 ; 𝑛 ∈ ℤ tan : + 180°𝑛 ; 𝑛 ∈ ℤ ⚫ Interval gegee: ⚫ Vind alle hoeke in interval deur substitusie 𝑛 = 1 , −1 , 2 , ens. 4.3 ⚫ sin 𝛼 = 3 cos 𝛼 ⚫ Deel deur cos 𝛼 om tan 𝛼 te kry ⚫ sin 𝛼 = cos 𝛽 ⚫ Gebruik die 90° reël vir sin ↔ cos 4.4 Faktorisering: bv. bv. bv. bv. sin 𝑥 + 3 sin 𝑥 cos 𝑥 = 0 sin2 𝑥 − 9cos2 𝑥 = 0 sin2 𝑥 − 2 sin 𝑥 cos 𝑥 + cos2 𝑥 = 0 6 cos 𝑥 − 3 cos2 𝑥 + 2 sin 𝑥 cos 𝑥 − 4 sin 𝑥 = 0 ⚫ ⚫ ⚫ ⚫ ⚫ Gemene faktor Verskil tussen kwadrate Drieterm Groepering Verskil en som van derdemagte EXERCISE: Mixed Trig questions 1. If cos 160° = 1 , determine, without a calculator, 𝑝 the value of sin 250° in terms of p. (4) 2. Prove the following identities: 1 2 1 1+cos 𝛼 2.1 (sin 𝛼 + tan 𝛼) = 1−cos 𝛼 2.2 1−cos 𝑥 sin 𝑥 = sin 𝑥 (5) (4) 1+cos 𝑥 3. If 1 − sin 𝐴 = 𝑝 and 1 + 𝑠𝑖𝑛 𝐴 = 𝑞, determine cos 2 𝐴 in terms of 𝑝 and 𝑞. 4. Solve : sin2 (𝜃 − 50°) = 1 4 (3) ; 0° ≤ 𝜃 ≤ 360° (6) 5. If sin 𝐴 = 𝑝 and cos 𝐵 = −√1 − 𝑝2 , where 𝑝 > 0 , 90° < 𝐴 < 360° and 180° < 𝐵 < 360° , prove that 𝐴 + 𝐵 = 360°. (5) 6. If 1 sin 𝜃 = 1 2𝑘 + 𝑘 2 and 90° ≤ 𝜃 ≤ 270° , determine, without a calculator, the value of 𝑐𝑜𝑠𝜃 in terms of 𝑘 , 𝑘 > 1. (6) 7. Simplify without a calculator : √4 sin 150° × 2 3 tan 225° (4) 8. Evaluate without a calculator: sin 237° cos 147° − cos 303° cos 213° tan 237° 9. If 𝛼 + 𝜃 = 90° and tan 𝛼 = 1 𝑘 (6) ;𝑘≠0, simplify in terms of 𝑘 : cos(𝜃−720°) sin2 (180°+𝛼).cos(90°−𝜃) 10.1 Prove the identity : cos 𝑥 1+sin 𝑥 (8) + 1+sin 𝑥 cos 𝑥 = 2 cos 𝑥 (5) 10.2 For which values of 𝑥 in the interval 0° ≤ 𝑥 ≤ 360° will the identity be undefined? (3) 11. Calculate the value of 𝐴 if 𝐴 = 2sin(𝑎−𝑚)° . sin(𝑏−𝑚)° . sin(𝑐−𝑚)°. (3) 12. Determine the general solution : 2 sin 𝑥 tan 𝑥 + tan 𝑥 = 2 sin 𝑥 + 1 . . . . sin(𝑧−𝑚)° (7) 13. Determine, without a calculator, the value of : 𝑠𝑖𝑛2 1° + 𝑠𝑖𝑛2 2° + 𝑠𝑖𝑛2 3°+ . . . +𝑠𝑖𝑛2 87° + 𝑠𝑖𝑛2 88° + 𝑠𝑖𝑛2 89° (4) 14. Simplify without a calculator : √3 sin 𝑥 sin2 72° + sin2 198° √3 cos(𝑥−90°) tan 120° sin 𝑥 (6) 15. Determine the general solution : √tan 𝜃 = 𝑥 + 1 if 𝑥 16. Determine the value of 𝑎 if cos 𝜃 = 17. If sin 𝜃 = calculate 5 𝑎−1 𝑎+1 1 𝑥 2 + 𝑥2 = 3 1 tan2 𝜃 (5) 1 + sin2 𝜃 in terms of ; 𝑎 ≠ −1 and 0° < 𝜃 < 90° , 90° < 𝜃 < 270° , 𝜃 = 90° + 𝛼 , 13 cos 𝛼 tan 𝜃 without a calculator. (6) (7) 18. Given : √5 cos 𝜃 − 2 = 0 Determine, without a calculator, the values of 𝑎 and 𝑏. (7) 𝑌 𝑃(𝑏; 4) • • 𝑀(6; 𝑎) 𝜃 𝑂 𝑋 19. 𝑃 and 𝑀 are points on a circle with centre 𝑂. 𝑅 is a point on the 𝑥 axis. If 𝑃𝑂̂𝑀 = 115°, calculate the values of 𝑎 and 𝑏. (6) 𝑌 𝑂 • 𝑃(4; 3) 𝑅 • • 𝑀(𝑎; 𝑏) 𝑋 20. In the diagram, 𝑃𝑂̂𝑄 = 90°. Determine the coordinates of 𝑄. (7) 𝑌 • 𝑃(𝑥; √3) 2 𝜃 𝛼 𝑂 𝑋 20 • 𝑄(𝑎; 𝑏) 21. 𝑂𝑃 = 2𝑅𝑂 and sin 𝛼 = 3 5 Determine, without a calculator, the coordinates of 𝑃. (6) 𝑌 • 𝑅(𝑥; 3) 𝛽 𝛼 𝑂 •𝑃 [138] • 𝑇 𝑋
