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Achievement objective M7-4

In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

  • apply trigonometric relationships, including the sine and cosine rules, in two and three dimensions.


  • Solves problems that can be modelled by trigonometric relationships.
  • Proves simple trigonometric identities making links to right-angled triangles.
  • Uses the area formula for triangles and the sine and cosine rules to solve problems.
  • Uses knowledge of right-angled triangles to find exact values of the trig ratios.
  • Uses radian measure.
  • Makes links between the unit circle M7-1, graphs of trig functions M7-2, and solutions of trig equations M7-7.
  • See key mathematical ideas on NZmaths.


M7-4 links from GM6-5, GM6-6 and to M8-6, M8-7.

Possible context elaborations

  • Arc length and area of sector problems:
    • semi-circular cross section trough problems
    • security light coverage using sectors
    • radar overlapping areas, search and rescue situations
    • alternating voltage applications from physics using unit circle.
  • Height measurement: Finding the height of an object
  • How far is it: Cosine rule
  • Navigation problem: Cosine rule
  • Sand: Finding the natural angle of inclines.
  • Sine rule introduction
  • Spherical shapes: Finding the angle between two points and the centre on a variety of balls.
  • Water sprinklers: Some water sprinklers continually turn left and right to water an area of lawn. The water lands on the ground a distance from the sprinkler. Work out the area of grass being watered.
  • Petrol barrel: A petrol barrel is lying on its side. How much petrol is left in the barrel? Draw a barrel end and include on it a volume scale.
  • Rotary clothesline: A rotary clothesline can have many shapes. Square, rectangle, pentagon or even hexagon. How much wire is needed for any one of these if they have 3 wires, 4 wires etc.
  • Use the length of your shadow to work out your height.
  • Calculate the angles on your bicycle frame.
  • Bolts: work out the distance that a bolt moves in a full turn. Compare several bolts. Compare the jack of a small car with the jack of a larger/older car. How does the thread differ?
  • A sharp pencil or pen has a tapered point. Measure distances to find the angle of the taper. Compare this with the taper of pencil sharpeners.
  • Work out the angle of a roof. Use this to find the height of the top of a roof.

Assessment for qualifications

NCEA achievement standards at level 1, 2 and 3 have been aligned to the New Zealand Curriculum. Please ensure that you are using the correct version of the standards by going to the NZQA website.

The following achievement standard(s) could assess learning outcomes from this AO:

Last updated September 17, 2018