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Sines, Cosines and Tangents

Sines, Cosines and Tangents. Of angles over 90 0. For Right Angled Triangles. 8cm. H. O. 4 cm. 30 0. Sin x = O / H. A. Sin 30 0 = 4 / 8 = 0.5. What about angles over 90 0 ?. What about angles over 90 0 ?. 30 0. Angles measured anti clockwise from horizontal. If angle is 150 0.

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Sines, Cosines and Tangents

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  1. Sines, Cosines and Tangents Of angles over 900

  2. For Right Angled Triangles 8cm H O 4 cm 300 Sin x = O/H A Sin 300 = 4/8 = 0.5

  3. What about angles over 900?

  4. What about angles over 900? 300 Angles measured anti clockwise from horizontal

  5. If angle is 1500 180 - 150 1500 Sin1500 = Sin300!!!!!!!!! Sin1500 = 0.5 300 300

  6. 300 and 1500 are related angles Anyone else in the family?

  7. 180 + 30 = 2100 What about angles over 1800? 300 210 is in the same family as 30 and 180 sin300 = 0.5, sin1500 = 0.5 - 0.5 Sin 2100 =

  8. 210 is more of a cousin! Any more relations?

  9. Related Angles so far 1500 300 360 – 30 = 3300 2100 3300 is the final relation

  10. But which side of the family is 330? Sin300 and Sin1500 = 0.5 Sin3300 = -0.5 Sin2100 = -0.5

  11. Finding all the relations There are 4 related angles (usually) Less than 900 Between 900 and 1800 a0 180 – a a0 360 - a 180 + a Between 1800 and 2700 Between 2700 and 3600

  12. (Neat Wee Diagram) 900 Introducing the NWD ii i Less than 900 Between 900 and 1800 a0 180 – a 00 1800 iv iii (3600) 180 + a 360 - a 4 Quadrants Between 1800 and 2700 Between 2700 and 3600 Each contains a related angle 2700

  13. i ii Easiest when starting in Quadrant 1 Finding relations using NWD a0 180 – a iv iii (Acute angle) 180 + a 360 - a i 700 Relations of 700 ii 180 – 70 = 1100 iii 180 + 70 = 2500 iv 360 – 70 = 2900

  14. Positive or Negative? Looking at the relations from earlier i ii Sin300 and Sin1500 = 0.5 iii iv Sin2100 and Sin330 = -0.5 Quadrants?

  15. i ii + + Adjusting the NWD a0 180 – a iv - - iii 180 + a 360 - a + i 500 Relations of 500 + ii 180 – 50 = 1300 - iii 180 + 50 = 2300 - iv 360 – 50 = 3100

  16. 0.5 How about cosine? Related AnglesCosine 600 1200 2400 3000 -0.5 -0.5 0.5 Cosine also has related angles i and iv are positive ii and iii are negative

  17. 1 And Tangent? Related AnglesTangent 450 1350 2250 3150 -1 1 -1 Tangent also has related angles i and iii are positive ii and iv are negative

  18. Sin Cos Sin Cos - S Only Sin + i A All + + ii + + Tan Tan - Further NWD Adjustment a0 180 – a + Sin Sin Cos Cos Only Tan + iv T - C - Only Cos+ iii + - Tan Tan 180 + a 360 - a - + Not so Neat! Sorted A bit better

  19. dair A S eldom i ii NWD Final Version a0 180 – a iv iii 180 + a 360 - a T alks C rap

  20. 0.64 0.77 0.84 0.64 -0.77 -0.84 (180 – 40) -0.64 - 0.77 0.84 (180 +40) -0.64 0.77 -0.84 (360 – 40)

  21. Relations from non acute angles Ex 2300 Quad iii 900 so 180 + a = 2300 a = 500 A S i ii 180 – a a0 can easily find rest of relations 00 1800 iv T C iii 180 – 50 = 1300 180 + a 360 - a 360 – 50 = 3100 2700

  22. Relations from non acute angles Ex 3300 Quad iv 900 so 360 - a = 3300 a = 300 A S i ii 180 – a a0 can easily find rest of relations 00 1800 iv T C iii 180 – 30 = 1500 180 + a 360 - a 180 + 30 = 2100 2700

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