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Level Five > Geometry and Measurement

Using Trigonometry

Achievement Objectives:

Achievement Objective: GM5-10: Apply trigonometric ratios and Pythagoras' theorem in two dimensions.
AO elaboration and other teaching resources


In this unit, students will explore the use of trigonometry to find unknown sides and angles in right-angled triangles, taking the concepts that were developed in the previous unit Introducing Trig and looking at different ways of practising the new skills. Students will then develop and reinforce skills to solve practical problems in trigonometry using the problem solving model

reality        arrow.       model     arrow.           solution    arrow.           check

It is important that good algebraic skills are used in setting out their work, and students can be encouraged to practice these individually and using cooperative logic problems in a group situation.

Specific Learning Outcomes: 
  • label right angled triangles with respect to a given angle
  • use trigonometric ratios to calculate the length of opposite and adjacent sides in right angled triangles
  • use trigonometric ratios to calculate the size of angles in right angled triangles

Maths skills required from other strands

  • algebraic skills for solving equations
  • rounding
Description of mathematics: 

This unit begins with an individual activity designed to reinforce the idea that the ratios determined in a measuring task are the same as those found by using the trigonometric ratios. The students need to measure the angle and label the triangles correctly before measuring the lengths of the sides and determining the ratios.

After demonstrating a suitable method for solving triangles involving how to choose the correct ratio, and how to solve the ensuing equation, students need to practice these skills both individually, and in a group situation using matching card activities and cooperative logic activities. Appropriate rounding needs to be discussed with the students and the reasonableness of answers needs to be looked at.

Applying trigonometry to practical situations is an important aspect of this unit, and suitable practical activities involving the students going outside to take measurements using long measuring tapes and clinometers (measure angles of inclination) will give the students a better understanding of the wide uses of trigonometry.

Required Resource Materials: 
rulers, large measuring tape, trundle wheels
compasses (or theodilite - measure horizontal and vertical angles)

Getting Started

  1. Each student needs a triangle worksheet and a recording sheet. To begin this activity the students need to learn how to label right-angled triangles with opp, adj and hyp in relation to a given angle. Once this has been done, a final measuring activity should be completed individually to reinforce the links between the ratios and sin, cos and tan for different angles.  Students should measure the angle size and lengths of sides for the ten triangles, and record the results on the recording sheet.  They can then calculate the values of sin, cos and tan for each triangle.  These results can be added to the three graphs created in the Introducing Trig to check for consistency of the results.
  1. Students should now be shown the way to use the sin, cos and tan ratios to:
    1. calculate unknown sides in right angled triangles, given an angle and another side.
    2. calculate unknown angles in right angles triangles, given two sides of the triangle.


  1. Working in pairs, each pair should be given a set of matching activities.  These matching activities can be modified to be used in a variety of ways.  The sheets can be cut up and placed in arbitrary order and the students challenged to put them in the correct order.  Alternatively the teacher can delete parts of the sheet and the students can be challenged to:
    1. write symbols to match words
    2. write words to match symbols
    3. fill in gaps in the words and symbols
  2. Working in groups of about 5, each group is given a cooperative logic problem to solve.
    1. Each person in the group is given one clue card.
    2. Each person reads out their clue to the rest of the group. (Some problems may have unnecessary information mixed in with facts).
    3. The group then works on solving the problem cooperatively.
    4. The solution should be checked by reading the clues again.


  1. Practical problem solving activities

The students can now use their trigonometric ability to challenge each other with practical problems.

Split the class into small groups and get each group to pose a question that could be solved using trigonometry.  Encourage them to find practical measuring tasks, where they measure distances, and angles to find the heights of buildings, trees, flagpoles, etc, or the width of streets, etc.

  1. Get the groups to swap problems and solve each other’s trigonometry questions.
UsingTrigonometryCM1.pdf51.75 KB
UsingTrigonometryCM2.pdf48.45 KB
UsingTrigonometryCM3.pdf116.91 KB
UsingTrigonometryCM4.pdf67.81 KB