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3D Shapes Names of 3D solids Prisms Volume Surface Area Plans and Elevations Isometric Drawing Units of measure Length, area and volume Capacity / Mass Metric and imperial measure Conversion / conversion graphs Real-life graphs Speed, distance time Parts of Circumference Area Circle geometry Equation of a circle Right-angled triangles Finding missing side lengths Angles and sine, cosine and tangent ratios KS3 Shape, Space and Measure Constructions Triangles Similar & Congruent Shapes Bisectors Loci Angles Angles on a straight line Angles at a point Parallel lines and transversals alternating / corresponding /opposite angles Supplementary angles Polygons: interior & exterior angles Extension work: Bearings Lines & Angles Line Segments Vertical / Horizontal Perpendicular / Parallel Types of angle Estimating measuring and drawing Direction of turn Compass directions Describing angles (90o= ¼ of turn) Pythagoras and Trigonometry Perimeter and Area Dimensions (including volume) Units of measure (including volume) Counting squares Intrinsic and Extrinsic information Rectangles, triangles and compound shapes Using Formulae (Rectangle, Triangle, Trapezium, Parallelogram) Circles Properties of 2D Shapes Names of polygons up to 10 sides Special Quadrilaterals and Triangles Geometric Properties Tessellation Transformations Basic congruent & similar shapes Coordinate geometry Reflection (include lines of symmetry) Rotation (include rotational symmetry) Translation (including vector notation) Enlargement All topics can be covered by the end of year 8. Shape 10: Pythagoras and Trigonometry Pythagoras’ Theorem and Trigonometry Must Should Identify the ‘hypotenuse’ on a right- Given the length of the hypotenuse angled triangle (regardless of the and one of the other sides, use orientation) Pythagoras’ Theorem to work out the length of the other side Know Pythagoras’ Theorem such that a2 + b2 = c2 for any right-angled Apply Pythagoras’ Theorems to triangle where c is the length of the problems involving area of a triangle hypotenuse where the height of the triangle is not Use Pythagoras’ Theorem to work out the length of the hypotenuse given the lengths of the other two sides Given an angle in a right-angled triangle, label the opposite and adjacent sides. Understand that trigonometry involves using or working out an angle in a right-angled triangle. Could Understand and give examples of the Pythagorean triplets Recognise that any triangle can be divided into two right-angled triangles, and that for an isosceles triangle, the two right-angled triangles will be congruent known Use the trigonometric functions to Understand trigonometric work out the length of a side given the functions by considering the unit length of one other side and an angle circle and by using Pythagoras’ Theorem Use trigonometric functions to work out the size of an angle given the lengths of two of the sides Learn the trigonometric functions; Sin = opposite ÷ hypotenuse Cos = adjacent ÷ hypotenuse Tan = opposite ÷ adjacent Key Words: right-angled triangle, hypotenuse, Pythagoras’ Theorem, length, area, Pythagorean Triplets, opposite, adjacent, trigonometry, trigonometric functions, sin, cos, tan, , unit circle, ratio Starters: Show a series of Pythagorean Triplets and ask students to investigate the ratio of the values Label sides of a right-angled triangle with the letters a, b and c or with the terms, opposite, adjacent and hypotenuse Activities: Make a ‘unit circle’ on a large enough scale for students to walk around and measure distances horizontally and vertically from the centre of the circle to experience the trigonometric ratios. Work out the areas of the three squares formed by drawing a right-angled triangle and using each length as the side length for each square / investigate the relationship between their areas. Plenaries: Learning framework questions: - When should we choose to use Trigonometry instead of Pythagoras’ Theorem? - How can you explain Pythagoras’ Theorem? - What do Sin and Cos tell us? - Where does Tan come from? - What are the Pythagorean Triplets? Shape 10: Pythagoras and Trigonometry Resources: Unit circle / graph paper 10 ticks worksheets Ratio sticks Possible Homeworks: Teaching Methods/Points: Pythagoras’ theorem Students must know that Pythagoras’ theorem applies to right-angled triangles; Given a right-angled triangle with lengths labelled a, b and c as follows; Where c is the length directly opposite the right angle a c This side is also called the ‘hypotenuse’ b Pythagoras’ theorem states that; a2 + b2 = c2 for any right-angled triangle. c2 a2 The theorem shows that if each length forms the side of a square, the area of the triangle with the side length of the hypotenuse is equal to the sum of the areas of the other two squares. b2 Students should be encouraged to identify the hypotenuse for right-angled triangles in a variety of different orientations where the right angle is not immediately obvious. Furthermore, students can investigate the Pythagorean triplets with side lengths in the ratio 3 : 4 : 5, showing that given lengths of 3n, 4n, and 5n (for the hypotenuse); (3n)2 + (4n)2 = (5n)2 hence 9n2 + 16n2 = 25n2 which, of course, will always be true. While it is relatively straight forward to use Pythagoras’ Theorem to work out the length of the hypotenuse given the lengths of the other two sides, students will need to practise rearranging the formula such that if; a2 + b2 = c2 then a2 = c2 – b2 and b2 = c2 – a2 (which is simple rearrangement on a maths table) Moreover, students must understand that, having substituted values for two of the lengths, and therefore having worked out the value of the square of one of the sides, the actual length of that side can be found by square rooting (√) (equivalent to asking the question … what value written two times as a product is equal to …) i.e. if a2 = 25, therefore a = 25 = 5. Encourage students to show every stage in their workings. Shape 10: Pythagoras and Trigonometry Examples for using Pythagoras’ Theorem to calculate missing lengths x 4 Pythagoras’ theorem; 42 + 32 = x2 16 + 9 = x2 25 = x2 5 = x (√ both sides) Pythagoras’ theorem; 72 + b2 = 8.52 49 + b2 = 72.25 b2 = 23.25 b =√23.25 = 4.82 (to 2d.p.s) 3 7 8.5 b To work out the height, h of the lighthouse; 30 metres Using Pythagoras’ theorem; h2 + 182 = 302 h2 + 324 = 900 h2 = 900 – 324 h = 576 h = 24 metres h 18 metres The Trigonometric Functions Students should understand at the most basic level that the trigonometric functions; sine, cosine and tangent apply to right-angled triangles. Whereas Pythagoras’ theorem allows you to find a missing length if you are given the other two lengths (it is all about lengths!), trigonometry allows you to find an angle, or use an angle to find a missing length if you are given just one other length. Ensure plenty of practice labelling the sides of the right-angled triangle with o, a and h, as shown below, given an angle, ; Angle, θ … the opposite length is opposite the given angle and the adjacent length is next to the given angle Hypotenuse h Opposite o Adjacent a And the trigonometric functions should be learned; Sine θ = o h Tan θ = o a Cos θ = a h “ Soh Toa Cah” or “Soh Cah Toa” may assist students in recalling the trig functions. Shape 10: Pythagoras and Trigonometry Understanding Trigonometry An effective way of introducing trigonometry is by creating a large unit circle outdoors, and using measuring tape to record the changing value of x and y as the angle of turn is increased in 5 or 10 degree increments. While this will not generate exact values, the relationship between the x coordinate and the y coordinate of each point of the circumference can be observed. The trigonometric functions are derived from the unit circle (circle of radius; 1 unit, centre; O) and the ratio of the opposite side compared to the adjacent side. Here is an example of the unit circle; 2 y-axis (x=0) 1 x-axis (y=0) -2 -1 1 2 -1 -2 This is the unit circle drawn on a coordinate grid. The radius of the circle is 1, hence the term, ‘unit circle’. The centre of the circle is (0,0) One point has been chosen on the circumference of the circle. The coordinate of the point happens to be; (0.8, 0.6). A right-angled triangle is drawn to this vertex. Remembering that, for a right-angled triangle, according to Pythagoras; a2 +b2 = c2 and given that a refers to the distance in the x direction and b refers to the distance in the y direction and c is equal to the radius of the circle; this gives us the equation for a circle; x2 + y2 = r2 [see Shape 9] Now, the angle formed by the hypotenuse as it meets the origin is given as θ (or “theta”). This is the angle measured at the centre of the circle in an anticlockwise direction where 0 degrees is equivalent to facing in the positive x direction. The trigonometric functions; cosine and sine, describe the changing values of the x and y coordinates respectively as the angle of turn, θ changes. 1 cosθ Sin θ θ for the unit circle; the distance in the x direction is called, cos θ and the distance in the y direction is called, sin θ and the hypotenuse will be equal to the radius of 1. Shape 10: Pythagoras and Trigonometry Where θ is zero degrees (0o), the distance in the x direction to the circumference of the unit circle must be 1, and the distance in the y direction must be 0. Similarly, where θ is 90o, the distance in the x direction to the circumference of the unit circle is 0, whilst the distance in the y direction is 1. Where θ is 45o, the coordinate at the circumference will be at a point where the distance in the x direction is equal to the distance in the y direction (i.e. along the line y = x) approximately at (0.707 , 0.707), although the exact value is expressed as; 2 . 2 Furthermore, due to the reflective properties of a circle, and the lines of symmetry naturally formed by the x and y axes, reflections of the right-angled triangle in the axes will have corresponding lengths. Therefore, Sine (sin) 30, for example, will be equal to Sine 150 and -Sine 210 and -Sine 330, as these all form rightangled triangles with corresponding angles of 30 degrees. This, of course, is also true for the Cosine (cos) of each angle such that Cos 30 = - cos 150 = - cos 210 = cos 330. This same symmetry occurs in the line, y = x. Hence, the Sine of 30 will equal the Cosine of 60. Similar circles, increasing the length of the hypotenuse, and the ratio of sides Thus far, the relationship of the trigonometric functions, sine and cosine, have been discussed in relation to each other and in relation to the amount of turn at the centre (θ) in the unit circle. Remembering that the unit circle has a radius of 1, and this is the length of the hypotenuse for a right-angled triangle formed in the unit circle; any scale factor enlargement of the circle will lead to a scale factor enlargement for every length of the triangle. This means that the ratio of the opposite or adjacent sides in comparison to the hypotenuse will remain the same. Students can understand this principle using ratio sticks … if the hypotenuse ratio stick is called, “one”, then the adjacent ratio stick is called, “one cos θ” and the opposite ratio stick is called, “one sin θ” … if the hypotenuse ratio stick is called, “two”, then the adjacent ratio stick is called, “two cos θ” and the opposite ratio stick is called, “two sin θ” … and so on! So, if Cos θ is the distance in the x direction, and Sin θ is the distance in the y direction, then for a circle, centre 0 and radius, h, the distance in the x direction becomes h Cos θ and the distance in the y direction becomes h Sin θ. As already explained, students must identify the opposite side, adjacent side and hypotenuse on the rightangled triangle based on given angle, θ; So a = h Cos θ and o = h Sin θ h o the distance in the y direction θ a the distance in the x direction Which leads to reasoning that; a Cos h and o Sin h Tan θ The tangent (tan) function is used to directly compare the distance in the y direction to the distance in the x direction. This function, therefore, describes the gradient of the hypotenuse and its changing value as θ, the angle of turn at centre O, changes. o sin o h So, tan cos a a h Shape 10: Pythagoras and Trigonometry Pythagoras’ Theorem and Trigonometry: Help Sheet Pythagoras’ theorem applies to right-angled triangles; a c Length c (directly opposite the right angle) is called the ‘hypotenuse’. b Pythagoras’ theorem states that; a2 + b2 = c2 for any right angle. By using this, and rearranging where necessary, it is possible to find the length of any side if you are given the lengths of the other two sides. Look at the following examples; x 4 Pythagoras’ theorem; 42 + 32 = x2 16 + 9 = x2 25 = x2 5 = x (√ both sides) Pythagoras’ theorem; 7 + b = 8.5 49 + b2 = 72.25 b2 = 23.25 b =√23.25 = 4.82 (to 2d.p.s) 3 7 2 2 2 8.5 b Trigonometry The trigonometric functions; sine, cosine and tangent apply to right-angled triangles. Whereas Pythagoras’ theorem allows you to find a missing length if you are given the other two lengths (it is all about lengths!), trigonometry allows you to find an angle, or use an angle to find a missing length if you are given just one other length. Angle, θ … the opposite length is opposite the given Hypotenuse angle and the adjacent length Opposite h is next to the given angle o Adjacent a Sine θ = o h Tan θ = o a Cos θ = a h Soh Toa Cah or Soh Cah Toa The trigonometric functions come from the unit circle (circle of radius; 1 unit, centre; O) and the ratio of the opposite side compared to the adjacent side. θ cosθ Sin θ 1 for the unit circle; the distance in the x direction is called, cos θ the distance in the y direction is called, sin θ the hypotenuse will be equal to the radius of 1. For similar circles, where the hypotenuse is multiplied by a value, h, the length of the adjacent side will be h cos θ and the opposite side will be h sin θ a Sine rule: for any triangle a b c SinA SinB SinC Shape 10: Pythagoras and Trigonometry B c Cosine rule: for any triangle A C b a 2 b 2 c 2 2bcCosA