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KS4 Mathematics

KS4 Mathematics. S3 Trigonometry. S3 Trigonometry. Contents. A. S3.2 The three trigonometric ratios. A. S3.3 Finding side lengths. A. S3.1 Right-angled triangles. S3.4 Finding angles. A. S3.5 Angles of elevation and depression. A. S3.6 Trigonometry in 3-D. A. Right-angled triangles.

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KS4 Mathematics

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  1. KS4 Mathematics S3 Trigonometry

  2. S3 Trigonometry Contents • A S3.2 The three trigonometric ratios • A S3.3 Finding side lengths • A S3.1 Right-angled triangles S3.4 Finding angles • A S3.5 Angles of elevation and depression • A S3.6 Trigonometry in 3-D • A

  3. Right-angled triangles A right-angled triangle contains a right angle. The longest side opposite the right angle is called the hypotenuse.

  4. The opposite and adjacent sides The two shorter sides of a right-angled triangle are named with respect to one of the acute angles. The side opposite the marked angle is called the opposite side. x The side between the marked angle and the right angle is called the adjacent side.

  5. Label the sides

  6. Similar right-angled triangles 10 cm 5 cm 6 cm 3 cm 37° 37° 4 cm 8 cm opp 6 6 8 opp adj 4 3 3 = = = = = = adj 8 10 10 hyp hyp 5 4 5 If two right-angled triangles have an acute angle of the same size they must be similar. For example, two triangles with an acute angle of 37° are similar. The ratio of the side lengths in each triangle is the same.

  7. Similar right-angled triangles

  8. S3 Trigonometry Contents S3.1 Right-angled triangles • A • A S3.3 Finding side lengths • A S3.2 The three trigonometric ratios S3.4 Finding angles • A S3.5 Angles of elevation and depression • A S3.6 Trigonometry in 3-D • A

  9. Trigonometry 12 cm 8 cm 6 cm ? 30° ? The word trigonometry comes from the Greek meaning ‘triangle measurement’. Trigonometry uses the fact that the side lengths of similar triangles are always in the same ratio to find unknown sides and angles. For example, when one of the angles in a right-angled triangle is 30° the side opposite this angle is always half the length of the hypotenuse. 4 cm 30°

  10. The sine ratio the length of the opposite side the length of the hypotenuse H We say: O P P O S I T E Y P O T opposite E sin θ= N U hypotenuse S E θ The ratio of is the sine ratio. The value of the sine ratio depends on the size of the angles in the triangle.

  11. The sine ratio In a right-angled triangle with an angle of 65°, what is the ratio of the opposite side to the hypotenuse? What is the value of sin 65°? This is the same as asking: To work this out we can accurately draw a right-angled triangle with a 65° angle and measure the lengths of the opposite side and the hypotenuse.

  12. The sine ratio opposite sin 65° = hypotenuse 11 cm 65° 10 = 11 10 cm What is the value of sin 65°? It doesn’t matter how big the triangle is because all right-angled triangles with an angle of 65° are similar. The length of the opposite side divided by the length of the hypotenuse will always be the same value as long as the angle is the same. In this triangle, = 0.91 (to 2 d.p.)

  13. The sine ratio using a table Here is an extract from a table of sine values: Angle in degrees .0 .1 .2 .3 .4 .5 63 0.891 0.892 0.893 0.893 0.894 0.895 64 0.899 0.900 0.900 0.901 0.902 0.903 65 0.906 0.907 0.908 0.909 0.909 0.910 66 0.914 0.914 0.915 0.916 0.916 0.917 What is the value of sin 65°? It is not practical to draw a diagram each time. Before the widespread use of scientific calculators, people would use a table of values to work this out.

  14. The sine ratio using a calculator 6 5 = sin What is the value of sin 65°? To find the value of sin 65° using a scientific calculator, start by making sure that your calculator is set to work in degrees. Key in: Your calculator should display 0.906307787 This is 0.906 to 3 significant figures.

  15. The cosine ratio the length of the adjacent side the length of the hypotenuse H We say, Y P O T E adjacent N cos θ= U S hypotenuse E θ A D J A C E N T The ratio of is the cosine ratio. The value of the cosine ratio depends on the size of the angles in the triangle.

  16. The cosine ratio In a right-angled triangle with an angle of 53°, what is the ratio of the adjacent side to the hypotenuse? What is the value of cos 53°? This is the same as asking: To work this out we can accurately draw a right-angled triangle with a 53° angle and measure the lengths of the adjacent side and the hypotenuse.

  17. The cosine ratio adjacent cos 53°= hypotenuse 10 cm 6 = 10 53° 6 cm What is the value of cos 53°? It doesn’t matter how big the triangle is because all right-angled triangles with an angle of 53° are similar. The length of the opposite side divided by the length of the hypotenuse will always be the same value as long as the angle is the same. In this triangle, = 0.6

  18. The cosine ratio using a table Here is an extract from a table of cosine values: Angle in degrees .0 .1 .2 .3 .4 .5 50 0.643 0.641 0.640 0.639 0.637 0.636 51 0.629 0.628 0.627 0.625 0.624 0.623 52 0.616 0.614 0.613 0.612 0.610 0.609 53 0.602 0.600 0.599 0.598 0.596 0.595 54 0.588 0.586 0.585 0.584 0.582 0.581 55 0.574 0.572 0.571 0.569 0.568 0.566 56 0.559 0.558 0.556 0.555 0.553 0.552 What is the value of cos 53°?

  19. The cosine ratio using a calculator 2 5 = cos What is the value of cos 25°? To find the value of cos 25° using a scientific calculator, start by making sure that your calculator is set to work in degrees. Key in: Your calculator should display 0.906307787 This is 0.906 to 3 significant figures.

  20. The tangent ratio the length of the opposite side the length of the adjacent side O P P O S I T E We say, opposite tan θ= adjacent θ A D J A C E N T The ratio of is the tangent ratio. The value of the tangent ratio depends on the size of the angles in the triangle.

  21. The tangent ratio In a right-angled triangle with an angle of 71°, what is the ratio of the opposite side to the adjacent side? What is the value of tan 71°? This is the same as asking: To work this out we can accurately draw a right-angled triangle with a 71° angle and measure the lengths of the opposite side and the adjacent side.

  22. The tangent ratio opposite tan 71° = 71° adjacent 4 cm 11.6 = 11.6 cm 4 What is the value of tan 71°? It doesn’t matter how big the triangle is because all right-angled triangles with an angle of 71° are similar. The length of the opposite side divided by the length of the adjacent side will always be the same value as long as the angle is the same. In this triangle, = 2.9

  23. The tangent ratio using a table Here is an extract from a table of tangent values: Angle in degrees .0 .1 .2 .3 .4 .5 70 2.75 2.76 2.78 2.79 2.81 2.82 71 2.90 2.92 2.94 2.95 2.97 2.99 72 3.08 3.10 3.11 3.13 3.15 3.17 73 3.27 3.29 3.31 3.33 3.35 3.38 74 3.49 3.51 3.53 3.56 3.58 3.61 75 3.73 3.76 3.78 3.81 3.84 3.87 76 4.01 4.04 4.07 4.10 4.13 4.17 What is the value of tan 71°?

  24. The tangent ratio using a calculator 7 1 = tan What is the value of tan 71°? To find the value of tan 71° using a scientific calculator, start by making sure that your calculator is set to work in degrees. Key in: Your calculator should display 2.904210878 This is 2.90 to 3 significant figures.

  25. Calculate the following ratios Use your calculator to find the following to 3 significant figures. 1) sin 79° = 0.982 2) cos 28° = 0.883 3) tan 65° = 2.14 4) cos 11° = 0.982 5) sin 34° = 0.559 6) tan 84° = 9.51 7) tan 49° = 1.15 8) sin 62° = 0.883 9) tan 6° = 0.105 10) cos = 0.559 56°

  26. The relationship between sine and cosine The sine of a given angle is equal to the cosine of the complement of that angle. sin θ = cos (90 – θ) cos (90 – θ)= a a 90 – θ b b sin θ = a a b b θ We can write this as, We can show this as follows,

  27. The three trigonometric ratios O P P O S I T E H Y P O T E N U S Opposite Adjacent Opposite θ E Cos θ= Sin θ= Tan θ= Hypotenuse Hypotenuse Adjacent A D J A C E N T Remember: S O H C A H T O A S O H C A H T O A

  28. S3 Trigonometry Contents S3.1 Right-angled triangles • A S3.2 The three trigonometric ratios • A • A S3.3 Finding side lengths S3.4 Finding angles • A S3.5 Angles of elevation and depression • A S3.6 Trigonometry in 3-D • A

  29. Finding side lengths opposite 12 cm x sin θ= hypotenuse x sin 56°= 56° 12 If we are given one side and one acute angle in a right-angled triangle we can use one of the three trigonometric ratios to find the lengths of other sides. For example, Find x to 2 decimal places. We are given the hypotenuse and we want to find the length of the side opposite the angle, so we use: x = 12 × sin 56° = 9.95 cm

  30. Finding side lengths adjacent cos θ= hypotenuse x 5 A 5 m ladder is resting against a wall. It makes an angle of 70° with the ground. What is the distance between the base of the ladder and the wall? We are given the hypotenuse and we want to find the length of the side adjacent to the angle, so we use: 5 m 70° x cos 70°= x = 5 × cos 70° = 1.71 m (to 2 d.p.)

  31. Finding side lengths

  32. S3 Trigonometry Contents S3.1 Right-angled triangles • A S3.2 The three trigonometric ratios • A S3.3 Finding side lengths • A S3.4 Finding angles • A S3.5 Angles of elevation and depression • A S3.6 Trigonometry in 3-D • A

  33. The inverse of sin sin–1 0.5 = sin 30° 0.5 sin–1 sin θ = 0.5, what is the value of θ? To work this out use the sin–1 key on the calculator. 30° sin–1 is the inverse of sin. It is sometimes called arcsin.

  34. The inverse of cos cos–1 0.5 = cos 60° 0.5 cos–1 Cos θ = 0.5, what is the value of θ? To work this out use the cos–1 key on the calculator. 60° Cos–1 is the inverse of cos. It is sometimes called arccos.

  35. The inverse of tan tan–1 1 = tan 45° 1 tan–1 tan θ = 1, what is the value of θ? To work this out use the tan–1 key on the calculator. 45° tan–1 is the inverse of tan. It is sometimes called arctan.

  36. Finding angles 8 cm 5 cm θ tan θ= opposite adjacent 8 tan θ= 5 Find θto 2 decimal places. We are given the lengths of the sides opposite and adjacent to the angle, so we use: θ = tan–1 (8 ÷ 5) = 57.99° (to 2 d.p.)

  37. Finding angles

  38. S3 Trigonometry Contents S3.1 Right-angled triangles • A S3.2 The three trigonometric ratios • A S3.3 Finding side lengths • A S3.5 Angles of elevation and depression S3.4 Finding angles • A • A S3.6 Trigonometry in 3-D • A

  39. Angles of elevation

  40. Angles of depression

  41. S3 Trigonometry Contents S3.1 Right-angled triangles • A S3.2 The three trigonometric ratios • A S3.3 Finding side lengths • A S3.6 Trigonometry in 3-D S3.4 Finding angles • A S3.5 Angles of elevation and depression • A • A

  42. Angles in a cuboid

  43. Lengths in a square-based pyramid

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