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NSW Syllabuses

Mathematics K–10 - Stage 5.2 - Measurement and Geometry Right-Angled Triangles (Trigonometry) ◊

Outcomes

A student:

  • MA5.2-1WM

    selects appropriate notations and conventions to communicate mathematical ideas and solutions

  • MA5.2-2WM

    interprets mathematical or real-life situations, systematically applying appropriate strategies to solve problems

  • MA5.2-13MG

    applies trigonometry to solve problems, including problems involving bearings

Content

  • Students:
  • Apply trigonometry to solve right-angled triangle problems (ACMMG224)
  • use a calculator to find the values of the trigonometric ratios, given angles measured in degrees and minutes ICT
  • use a calculator to find the size in degrees and minutes of an angle, given a trigonometric ratio for the angle ICT
  • find the lengths of unknown sides in right-angled triangles where the given angle is measured in degrees and minutes
  • find the size in degrees and minutes of unknown angles in right-angled triangles
  • solve a variety of practical problems involving angles of elevation and depression, including problems for which a diagram is not provided
  • draw diagrams to assist in solving practical problems involving angles of elevation and depression (Communicating, Problem Solving) CCT
  • interpret three-figure bearings (eg 035°, 225°) and compass bearings (eg SSW) L
  • interpret directions given as bearings and represent them in diagrammatic form (Communicating, Reasoning) CCT
  • solve a variety of practical problems involving bearings, including problems for which a diagram is not provided
  • draw diagrams to assist in solving practical problems involving bearings (Communicating, Problem Solving) CCT
  • check the reasonableness of solutions to problems involving bearings (Problem Solving) CCT

Background Information

When setting out a solution to a problem that involves finding an unknown side length or angle in a right-angled triangle, students should be advised to give a simplified exact answer, eg \( 25 \sin 42^{\circ} \) metres or \( \sin A = \frac{4}{7}\), and then to give an approximation correct to a given, or appropriate if not given, level of accuracy.

Students could be given practical experiences in using clinometers for finding angles of elevation and depression and in using magnetic compasses for bearings. They need to recognise the 16 points of a mariner's compass (eg SSW) for comprehension of compass bearings in everyday life, eg weather reports.

Students studying circle geometry will be able to apply their knowledge, skills and understanding in trigonometry to many problems, making use of the right angle between a chord and a radius bisecting the chord, between a tangent and a radius drawn to the point of contact of the tangent, and in a semicircle.

Language

Students need to be able to interpret a variety of phrases involving bearings, such as:

The bearing of Melbourne from Sydney is 230°

A plane flies to Melbourne on a bearing of 230°

from Sydney A plane flies from Sydney to Melbourne on a bearing of 230°

A plane leaves from Sydney and flies on a bearing of 230° to Melbourne.

Students should be taught explicitly how to identify the location from where a bearing is measured and to draw the centre of the compass rose at this location on a diagram. In each of the examples above, the word 'from' indicates that the bearing has been measured in Sydney and, consequently, in a diagram, the centre of the relevant compass rose is at Sydney.

To help students understand questions that reference a path involving more than one bearing, they may need to be explicitly shown to look for words, such as 'after this', 'then' and 'changes direction', that indicate a change of bearing. A new compass rose needs to be centred on the location of each change in direction.