Exact values
- 2. What is to be Learned? • The exact trig values of vital angles • How to use exact values with related angles
- 3. Finding the angles 450 450 1 1 √2 H O A sin450 = 1 /√2 cos450 = 1 /√2 tan450 = 1 /1 = 1
- 4. Finding the angles 600 600 2 2 2 H O A sin600 = √3 /2 cos600 = 1 /2 tan600 = √3 /1 = √3 600 1 1 300 300 √3
- 5. Finding the angles 600 2 H O A sin600 = √3 /2 cos600 = 1 /2 tan600 = √3 /1 = √3 1 300 √3 cos300 = √3 /2 sin300 = 1 /2 tan300 = 1 /√3
- 6. And Finally y = sinx y = cosx Sin00 = 0 Sin900 = 1 Cos00 = 1 Cos900 = 0
- 7. Exact Values There are some trig values we must memorise – they are exact (not approximations)
- 8. 0000 303000 454500 606000 909000 sin cos tan 0 π /2 π /3 π /4 π /6 degrees rads 0 1½ 1 /√2 √3 /2 1 √3 /2 1 /√2 ½ 0 0 1 /√3 1 √3 ∞
- 9. Finding all the relations There are 4 related angles (usually) Less than 900 Between 900 and 1800 Between 1800 and 2700 Between 2700 and 3600 a0 a0 180 – a 180 + a 360 – a 3200 1400 2200 400
- 10. Finding relations using NWD a0 180 – a 180 + a 360 – a iii iii iv Easiest when starting in Quadrant 1 (Acute angle) Relations of 700 i 700 ii 180 – 70 = 1100 iii 180 + 70 = 2500 iv 360 – 70 = 2900 AS T C
- 11. Relations from non acute angles Ex 2300 180 – a 180 + a iii iii iv a0 360 - a AS T C Quad iii so 180 + a = 2300 a = 500 can easily find rest of relations 180 – 50 = 1300 360 – 50 = 3100 0, 3600 1800 2700 - need related acute angle 900
- 12. Relations from non acute angles Ex 3300 180 – a 180 + a iii iii iv a0 360 – a AS T C Quad iv so 360 – a = 3300 a = 300 rest of relations 180 – 30 = 1500 180 + 30 = 2100 0, 3600 900 1800 2700 900
- 13. Find relations of these angles 1. 1000 800 2600 2800 2. 2000 200 1600 3400 3. 3000 600 1200 2400 4. 1950 150 1650 3450 5. 3080 520 1280 2320
- 14. Exact Values of non acute angles Ex sin 3300 180 – a 180 + a iii iii iv a0 360 – a AS T C Quad iv so 360 – a a = 300 sin 300 sin3300 = ½ - need related acute angle – = 3300 900 1800 0 Acute Exact Value = ½ Related Exact Value Check Sign 00 ,3600
- 15. Exact Values of non acute angles Ex tan 1350 180 – a 180 + a iii iii iv a0 360 – a AS T C Quad ii so 180 – a a = 450 tan 450 tan1350 = 1 - need related acute angle – = 1350 900 1800 0 Acute Exact Value = 1 Related Exact Value Check Sign 00 ,3600
- 16. Exact Values of non acute angles Ex cos 2400 180 – a 180 + a iii iii iv a0 360 – a AS T C Quad iii so 180 + a a = 600 cos 600 cos 2400 = ½ - need related acute angle – = 2400 900 1800 0 00 ,3600 Acute Exact Value = ½ Related Exact Value Check Sign
- 17. You Try 1. sin1200 2. cos2400 3. tan1350 4. sin2250 5. tan2100 6. cos3300 7. sin1800 (beware!) √3 /2 - ½ -1 - 1 /√2 1 /√3 √3 /2 0 Key Question
- 18. It gets nastier! Ex cos5π /4 180 – a 180 + a iii iii iv a0 360 - a AS T C Quad iii so 180 + a = 2250 a = 450 cos450 = 1 /√2 cos2250 = - 1 /√2 55π /4 X 180 / π = 900 π /4 π = 2250 i.e. coscos55π /4 = - 1 /√2