C2 st lecture 8 pythagoras and trigonometry handout

download C2 st lecture 8   pythagoras and trigonometry handout

of 41

  • date post

    11-May-2015
  • Category

    Education

  • view

    113
  • download

    2

Embed Size (px)

Transcript of C2 st lecture 8 pythagoras and trigonometry handout

  • 1.Lecture 8 - Pythagoras and Trigonometry C2 Foundation Mathematics (Standard Track) Dr Linda Stringer Dr Simon Craik l.stringer@uea.ac.uk s.craik@uea.ac.uk INTO City/UEA London

2. Triangles A triangle has 3 sides and 3 internal angles. The sum of the internal angles is 180 (degrees). + + = 180 3. Types of triangle If all three sides are the same length it is called an equilateral triangle. In an equilateral triangle all of the angles are 60. If only two sides are of the same length it is called an isosceles triangle. In an isoceles triangle, two of the angles are equal. If no two sides are of the same length it is called a scalene triangle. 4 cm 4 cm 4 cm 60 60 60 6 cm 6 cm 80 80 4. Right-angled triangles hypotenuse An angle of 90 is a right-angle. If one of the angles in a triangle is 90 then we call it a right-angled triangle. In a right-angled triangle, the side opposite the right-angle is called the hypotenuse. It is always the longest side. 5. Pythagoras theorem Pythagoras theorem states that in a right-angled triangle a2 + b2 = hyp2 a b hyp The square of the hypotenuse is equal to the sum of the squares of the other two sides. 6. Pythagoras Pythagoras theorem is attributed to Pythagoras, a Greek mathematician/philosopher in 6th century B.C. Evidence suggests it was known to the Chinese and Babylonians well before this 7. Example Consider the following right-angled triangle 3 4 c What is the length c (the hypotenuse)? c2 = 32 + 42 = 9 + 16 = 25 c = 25 = 5 8. Example - your turn Consider the following right-angled triangle 1 2 c What is the length c? Give your answer in surd form 9. Example - your turn Consider the following right-angled triangle 3 7 c What is the length c? Give your answer to 1 decimal place 10. Example Consider the following right-angled triangle 12 b 13 What is the length b? 132 = 122 + b2 169 = 144 + b2 25 = b2 b = 25 = 5 11. Example - your turn Consider the following right-angled triangle 5 b 8 What is the length b? Give your answer in surd form 12. Proof of Pythagoras theorem Take a right angled triangle: a b c B A As it is a right-angled triangle A + B = 90. 13. Proof of Pythagoras theorem Make 4 copies of the triangle and arrange into a square. a b c B A a b c B A a b c B A a b c B A The central area is a square as A + B = 90. The centre square has area c2. 14. Proof of Pythagoras theorem Rotate the triangle to form 2 rectangles. a b c B A a b c B A A B B A a a b b The area outside the triangles remains the same. However, it is now made up of 2 squares. One is of area a2 and the other is of area b2. a2 + b2 = c2 15. The trigonometric ratios: Sine, Cosine and Tangent Take a right angled triangle: adjacent opposite hypotenuse A We dene three functions: sin(A) = opp hyp , cos(A) = adj hyp , tan(A) = opp adj 16. Your turn Use your calculator to nd the following ratios. For trigonometric functions, always use surd form or 3 d.p. sin(10) sin(20) sin(30) sin(40) sin(90) 17. Example Consider the following triangle b 8 35 What is the length b? sin(35) = opp hyp = b 8 b = 8 sin(35) = 4.6 to 1 d.p. 18. Example Consider the following triangle b 4 30 What is the length b? cos(30) = adj hyp = b 4 b = 4 cos(30) = 2 3 19. Your turn Use your calculator to nd the following angles. Give your answers to 1 d.p. sin1 (0.760) sin1 (1 2 ) cos1(0) cos1( 3 2 ) tan1 (1) tan1 (.5) 20. Example Consider the following triangle 2 7 A What is sin(A) ? sin(A) = opp hyp = 2 7 What is angle A ? Give your answer to 1 d.p. A = sin1 (2 7 ) = 16.6 21. Example Consider the following triangle 3 6 A What is the angle A? sin(A) = opp hyp = 3 6 = 1 2 A = sin1 (1 2 ) = 30 22. A trigonometric identity If A is any angle, then cos2 (A) + sin2 (A) = 1 Proof Consider an angle A. Build a right-angled triangle with angle A. a b c A We have cos(A) = b c and sin(A) = a c It follows that cos2(A) + sin2 (A) = b c 2 + a c 2 = b2+a2 c2 = 1 (by Pythagoras Theorem) 23. Sine rule In any triangle we have the following: a sin(A) = b sin(B) = c sin(C) a bc B C A 24. Example Find the length of side A a sin(A) = b sin(B) a sin(60) = 7 sin(40) a = 7 sin(40) sin(60) = 9.4 to 1 d.p. a 7 40 60 25. Your turn Find the length of side A a sin(A) = b sin(B) a 12 36 72 26. Proof of the sine rule Draw a line down from the corner A down to the edge BC so it is at right angles. sin(B) = AP c sin(C) = AP b a bc B C A P Then c sin(B) = AP = b sin(C) so b sin(B) = c sin(C) 27. Example Two polar bears are at the North pole. They are 100m away from each other and directly between them is the north pole. One polar bear has an angle of 20 to the top of the pole and the other 30. How tall is the pole? 100m 20 30 28. Example We have the triangle 100 bc 20 30 A P 29. Example 100 bc 20 30 A P The angle at A is 180 20 30 = 130. We have c sin(30) = 100 sin(130) , so c = 100 sin(30) sin(130) = 65.3 Then sin(20) = AP 65.3 , so AP = 65.3 sin(20) = 22.3 30. Cosine rule In any triangle we have the following: a2 = b2 + c2 2bc cos(A) a bc B C A 31. Example Find the angle A 4 56 A 42 = 52 + 62 2 5 6 cos(A) 16 = 25 + 36 60 cos(A) 45 = 60 cos(A) 3 4 = cos(A) A = cos1(3 4 ) = 41.4 32. Proof of the cosine rule Draw a line down from the corner A down to the edge BC so it is at right angles. We have cos(B) = BP c cos(C) = PC b a bc B C A P Then a = BP + PC = c cos(B) + b cos(C) so a2 = ac cos(B) + ab cos(C) 33. Proof of the cosine rule By symmetry a2 = ac cos(B) + ab cos(C) b2 = ba cos(C) + bc cos(A) c2 = ca cos(B) + cb cos(A) Then b2 + c2 = ba cos(C) + bc cos(A) + ca cos(B) + cb cos(A) = a2 + 2bc cos(A) We rearrange to get: a2 = b2 + c2 2bc cos(A) 34. Area of a right-angled triangle In a right-angled triangle a b Area= 1 2 ab 35. Example What is the area of the triangle? 6 b 10 102 = 62 + b2 b2 = 64 b = 8 Area = 1 2 8 6 = 24 36. Proof of the area of a right-angled triangle Take two copies of the right-angled triangle They form a rectangle with area ab. Hence, the area of one triangle is 1 2 ab 37. Area of a triangle In any triangle we have the following: Area = 1 2 ab sin(C) a bc B C A 38. Example What is the area of the triangle? 5 46 A 52 = 42 + 62 2 4 6 cos(A) 25 = 52 48 cos(A) 27 = 48 cos(A) 9 16 = cos(A) A = cos1( 9 16 ) = 55.8 Area = 1 2 4 6 sin(55.8) = 9.9 to 1 d.p. 39. Proof of the area of a triangle Draw a line down from the corner A down to the edge BC so it is at right angles. The area of triangle ACP is 1 2 CP PA The area of triangle ABP is 1 2 PB PA a bc B C A P Then the total area is (1 2 CP PA) + (1 2 PB PA) As CP + PB = a, we have Area = 1 2 a PA Then since sin(C) = PA b , it follows that Area = 1 2 ab sin(C) 40. Formulae Pythagoras Theorem a2 + b2 = hyp2 Trigonometric ratios sin(A) = opp hyp , cos(A) = adj hyp , tan(A) = opp adj Sine rule a sin(A) = b sin(B) = c sin(C) Cosine rule a2 = b2 + c2 2bc cos(A) Area Area = 1 2 ab sin(C) Pythagoras theorem and the trigonometric ratios apply to right-angled triangles only 41. Practise Question sheet 8 Course book Attwood et al (2008) Edexcel AS and A Level Modular Mathematics, Core Mathematics 1. Heinemann, Pearsons Other recommended books Croft and Davison. Foundation Mathematics. 5th edition. Pearson Prentice Hall. Lakin, S. How To Improve your Maths Skills. Pearson. http://www.mathcentre.ac.uk/resources/uploaded/mc-ty- trigratios-2009-1.pdf mathcentre.ac.uk khanacademy.org bbc bitesize standard grade http://www.cliffsnotes.com/math/trigonometry Make up your own triangles