Achievement objective M74
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.
Indicators
 Solves problems that can be modelled by trigonometric relationships.
 Proves simple trigonometric identities making links to rightangled triangles.
 Uses the area formula for triangles and the sine and cosine rules to solve problems.
 Uses knowledge of rightangled triangles to find exact values of the trig ratios.
 Uses radian measure.
 Makes links between the unit circle
M71, graphs of trig functions
M72, and solutions of trig equations
M77.
 See
key mathematical ideas on NZmaths.
Progression
M74 links from GM65, GM66 and to M86, M87.
What is new/changed?
 This was previously located in the geometry strand at level 6.
Possible context elaborations
 Sample identities include: sin2x + cos2x = 1,
,
.

Sample exact values could include:
.
 Arc length and area of sector problems:
 semicircular 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:
 AS91259 Mathematics and statistics 2.4 Apply trigonometric relationships in solving problems
Refer to the
mathematics and statistics matrix.
Last updated September 26, 2013
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