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Further Mathematics

Further Mathematics. Geometry & Trigonometry Summary. Introduction. In this lesson we will consider how we can choose the right technique to use for a given problem. This will include… Things to do when starting a question Choosing the right technique Things to check before you finish.

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Further Mathematics

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  1. Further Mathematics Geometry & Trigonometry Summary

  2. Introduction In this lesson we will consider how we can choose the right technique to use for a given problem. This will include… • Things to do when starting a question • Choosing the right technique • Things to check before you finish

  3. 1.Starting a question • Read the question carefully. • Draw a diagram and list any values that have been given. • Add any extra information that can be easily worked out using geometry laws • Eg: If you have two angles in a triangle find the third (180° – other two angles). • Convert from bearings to angles • Double check the question for more information • Eg: for similar figures, which one is the original

  4. 2. Choosing the right approach To get started we will divide all of the possible questions into five groups. • Problems involving perimeters • Problems involving areas • Problems involving volumes • Problems involving similar figures • Problems involving lengths and angles of triangles

  5. 2.1 Problems involving perimeter • Find the total distance around the outside of the shape. • For questions involving circles useC = 2πr

  6. 2.2 Problems involving area • Simple shapes • Choose from the formulas on p332 • Composite shapes • Divide the shape into simple shapes • Total Surface Area of a 3D shape • For common shapes choose from the formulas on p338 • For other shapes draw a net and add the areas of each face (p339) • For triangles where base and height are not known • For problems involving Area, 2 sides, 1 angle useArea = ½ ab sin C • For problems involving Area, 3 sides use Heron’s Formula (see page 422)

  7. 2.3 Problems involving volume • Prisms • Use Vprism= Area of cross section  height • Pyramids & Cones • Use Vpyramid = 1/3 Area of base  height • Spheres • Use Vsphere = 4/3πr3 • Composite shapes • Divide the shape into prisms, pyramids & cones and spheres. Find the volume of each and add them to get the total.

  8. Examples • Find the perimeter of this shape. • Find the area.

  9. Examples • Find the total surface area. • Find the volume.

  10. Examples • Find the area. • Find the area.

  11. 2.4 Problems involving similar figures • Proving similarity • Use AAA, SSS (or for similar triangles SAS) • Finding the scale factor • Use k = length on copy ÷ length on original • Finding lengths using k • Use the ratios of corresponding sides or • Use the scale factor (above). • Problems involving areas and volumes • Use lsf = k, asf = k2 and vsf = k3

  12. 2.5 Problems involving lengths and angles of triangles • Right angled triangles • For problems involving 3 sides use Pythagoras theorem • For problems involving 2 sides and 1 angle use Trigonometric ratios (SOHCAHTOA) • Triangles that do not have a right angle • For problems involving 2 sides, 2 angles use the Sine rule. • To find an obtuse angle useobtuse angle = 180° - acute angle • For problems involving 3 sides, 1 angle use the Cosine rule. • To find an unknown side: • To find an unknown angle:

  13. Examples • What is the angle at B? • What is the angle s?

  14. Examples • What is the angle of elevation? • What is the length of the unknown side?

  15. 3. Before you finish • Don’t forget the last step in the calculation • Did you need to take the square root? • Did you need to use an inverse trig function (sin-1, cos-1 or tan-1) • Have you shown the correct units? • Have you used the right number of decimal places? • If the answer was an angle… • Should it be converted to a bearing? • Should it be in degrees and minutes? • Have you answered the question?

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