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PROGRAMME F8

PROGRAMME F8. TRIGONOMETRY. Programme F8: Trigonometry. Angles Trigonometric identities Trigonometric formulas NB: I have slightly edited the book’s slides. Programme F8: Trigonometry. Angles Rotation Radians Triangles Trigonometric ratios Reciprocal ratios Pythagoras’ theorem

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PROGRAMME F8

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  1. PROGRAMME F8 TRIGONOMETRY

  2. Programme F8: Trigonometry Angles Trigonometric identities Trigonometric formulas NB: I have slightly edited the book’s slides

  3. Programme F8: Trigonometry Angles Rotation Radians Triangles Trigonometric ratios Reciprocal ratios Pythagoras’ theorem Special triangles

  4. Programme F8: Trigonometry Angles Rotation When a straight line is rotated about a point it sweeps out an angle that can be measured in degrees or radians A straight line rotating through a full angle and returning to its starting point is said to have rotated through 360 degrees (360o ) One degree = 60 minutes (60'), and one minute = 60 seconds (60'')

  5. Programme F8: Trigonometry Angles: Radians When a straight line of length r is rotated about one end so that the other end describes an arc of length r the line is said to have rotated through 1 radian – 1 rad Since circumference has total length 2 πr, there are 2π radians in a full circle. So 1 radian is 360/ 2π degrees, i.e. about 57 degrees.

  6. Useful Numbers of Radians and Degrees 2 π radians = 360 degrees π radians = 180 degrees π/2 radians = 90 degrees π/3 radians = 60 degrees π/4 radians = 45 degrees π/6 radians = 30 degrees

  7. All at Sea (added by John Barnden) The circumference of the Earth is about 24,900 miles. That corresponds to 360 x 60 minutes of arc, = 21,600' So 1' takes you about 24,900/21,600 miles = about 1.15 miles. A nautical mile was originally defined as being the distance that one minute of arc takes you on any meridian (= line of longitude). This distance varies a bit as you go along the meridian, because of the irregular shape of the Earth. A nautical mile is now defined as 1852 metres, which is about 1.15 miles. A knot is one nautical mile per hour. NB: 60 knots is nearly 70 miles/hour. Look up nautical miles and knots on the web – it’s interesting.

  8. Programme F8: Trigonometry Angles Triangles All triangles possess shape and size. The shape of a triangle is governed by the three angles and the size by the lengths of the three sides

  9. Programme F8: Trigonometry Angles Trigonometric ratios

  10. Programme F8: Trigonometry Trigonometric ratios: in a right-angled triangle AB = the “hypotenuse” ERROR in tangent formula: should be AC/BC !!!!

  11. Programme F8: Trigonometry Reciprocal ratios

  12. Programme F8: Trigonometry Pythagoras’ s Theorem The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides

  13. Programme F8: Trigonometry Special triangles Right-angled isosceles

  14. Programme F8: Trigonometry Special triangles, contd Half equilateral

  15. Programme F8: Trigonometry Angles Special triangles Half equilateral

  16. Programme F8: Trigonometry Angles Trigonometric identities Trigonometric formulas

  17. Programme F8: Trigonometry Trigonometric identities The fundamental identity Two more identities Identities for compound angles

  18. Programme F8: Trigonometry The fundamental identity The fundamental trigonometric identity is derived from Pythagoras’ theorem

  19. Programme F8: Trigonometry Two more identities Dividing the fundamental identity by cos2

  20. Programme F8: Trigonometry Last one: Dividing the fundamental identity by sin2

  21. Beyond Pythagoras (added by John Barnden) Cosine Rule Ignore the outer triangle. Let the sides of the inner triangle ABC have lengths a, b, c (opposite the angles A, B, C, respectively). Then: c2 = a2 + b2 – 2ab.cos C This works for any shape of triangle. When C = 90 degrees, we just get Pythagoras, as cos 90o = 0. EX: What happens when C is zero?

  22. Beyond Pythagoras, contd The result on the previous slide can easily be shown be dropping a perpendicular from vertex A to line BC. Try it as an EXERCISE. Use Pythagoras on each of the resulting right-angle triangles. You’ll also need to use the Fundamental Identity.

  23. Another Interesting Fact (added by John Barnden) Sine Rule a/sin A = b/sin B = c/sin C This again can easily be seen by dropping a perpendicular from any vertex to the opposite side. Try it as an EXERCISE. Just use the definition of sine twice to get two different expressions for the length of the perpendicular.

  24. Programme F10: Functions Switching to Programme F10 briefly

  25. Programme F10: Functions Trigonometric functions Rotation For angles greater than zero and less than /2 radians the trigonometric ratios are well defined and can be related to the rotation of the radius of a unit circle:

  26. Programme F10: Functions Trigonometric functions: Rotation beyond 90 degrees By continuing to rotate the radius of a unit circle the trigonometric ratios can extended into the trigonometric functions, valid for any angle. Take AB positive if above the line, negative if below.Take OB positive if to the right, negative to the left. So AB positive, OB negative in case shown below. Hypotenuse (the radius) always has positive length.

  27. Programme F10: Functions Trigonometric functions Rotation The sine function:

  28. Programme F10: Functions Trigonometric functions Rotation The cosine function:

  29. Programme F10: Functions Trigonometric functions The tangent The tangent is the ratio of the sine to the cosine:

  30. Programme F8: Trigonometry Switching back to Programme F8

  31. Programme F8: Trigonometry Angles Trigonometric identities Trigonometric formulas

  32. Programme F8: Trigonometry Trigonometric formulas Sums and differences of angles Double angles Sums and differences of ratios Products of ratios

  33. Programme F8: Trigonometry Trigonometric formulas Sums and differences of angles (NB: there’s a typo on LHS of 2nd sine formula – should have a minus sign instead of a plus sign – John B.)

  34. Programme F8: Trigonometry Double angles

  35. Programme F8: Trigonometry Sums and differences of trig functions

  36. Programme F8: Trigonometry Products of trig functions

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