Trigonometry Trouble
The intention of Trigonometry Trouble is to cause a few troubles for students who will need to
problem solve to complete the task. This task is used to introduce students to trigonometry by
focussing solely on the tangent ratio. Prior knowledge of gradient is required.
Mathematical Focus
The tan ratio
MA5.1-10MG A student applies trigonometry, given diagrams, to solve problems, including
problems involving angles of elevation and depression
MA5.2-13MG A student applies trigonometry to solve problems, including problems involving
bearings
Groundwork
Make enough copies of the triangle and table templates.
For the clinometer students will need a copy of the protractor, sticky tape, 15cm of string, 2
paperclips, a drinking straw and a tape measure.
Students will need rulers
Timing 90-120 minutes
The Task 1
Students are to cut out the six triangles and sort them into sets of similar triangles. They then line
them up in the first quadrant of the Cartesian Planes so the blue dots sit on top of each other above
the origin, and the adjacent sides are on the x-axis.
Question
What do you notice? What “lines up” or matches?
How would you describe the gradient made by the hypotenuse? What are the rise and the
run in these triangles?
Encourage the use of the terms hypotenuse, adjacent side, opposite side and similar.
Introduce the idea that when we are talking about right angle triangles the special word for the
gradient made by the hypotenuse is the tangent ratio. The formula is:
𝑟𝑖𝑠𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
tan  = 𝑟𝑢𝑛 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
when the angle  is the angle made by the line and the positive horizontal axis.
Questions
Tan 600 is the ratio of two side lengths. The side opposite the 600 angle of a right angled triangle
is divided by the adjacent side. Look at the three similar triangles with 600 angles. Look
especially at the 600 angle. Which of the triangles has the greatest value for the ratio tan600?
Students often treat trigonometric ratios as if they are determined by the size of the triangle, or the
length of one of the sides. The question above provides the teacher with a measure of how well
students understand this at this stage. To understand ratios requires proportional reasoning.
Proportional reasoning impacts understanding in a range of Stage 4 substrands including fractions,
percentages, similarity, algebra and indices.
Have students measure the opposite and adjacent sides of each of the three right angled triangles
with 600 angles. Students use their measurements to calculate the tan ratio for each of the three
triangles. Answers can be compared and small differences explained by inaccuracies of
measurement.
The Task 2
Display the first row of the table from the Changes document and the triangle ABC.
Angle A
Increases
Angle B
Angle C
Side length
AB
Side length
BC
Fixed
Side Length
AC
Fixed
Tan A
Questions
Visualise the change described in the table. Angle A increases while the right angle stays fixed
at 900, side AC remains fixed.
Discuss with your partner what happens to angle B.
Asks whether any pairs could not agree. Take a quick survey of the student pairs as to whether
angle B increases, decreases or stays the same.
Demonstrate with change <B file. http://tube.geogebra.org/m/1400143
Lift point B upwards to increase angle A.
What happens to the sides AB and BC?
Distribute the Changes sheet and have students complete the top row immediately and consider
how to complete the table row by row in pairs.
Reflection 2
Facilitate a class discussion by working through student responses row by row. It is not intended that
the answers are marked. It is an opportunity for students to engage in reasoning, listening and
visualisation.
The Task 3
The clinometers take about 15 minutes to assemble. Students can work in pairs or groups of three.
A protractor template is here (Protractor doc). A hole punch or scissors are needed to help tie the
sting to the protractor
630
Eye height
Identify a building, statue or tree on the school grounds. The surrounding ground needs to be
relatively flat. This will be discussed in the reflection stage.
Each group of students will measure from different points. The angle between the vertical and the
line of sight to the top of the structure is measured with the clinometer.
Student names
Eye height
Distance to
structure base
Angle to the vertical
As will be obvious to teachers, the next step is to calculate the height of the structure using the tan
ratio. For most students the process will need to be carefully scaffolded. Some important steps are
below.
1. Draw a labelled triangle represented by the three points
a. Student eye position
b. Point on the structure level with student eye height
c. The top of the structure
2. Mark in the vertical angle measured.
Students will often find this difficult. They expect the angle to be at the point where they
were standing. It may be necessary for students to replicate how they held their clinometers
to measure the angle outside. Ask which direction the string hangs. Establish that it is
vertical. Ask where the straw is on the diagram. Which side is it sitting on? Use the diagram
above, or similar, to determine where the angle is positioned in the triangle.
3. Mark in the distance from the point where they stood to the bottom of the structure.
4. Label the sides as opposite, adjacent and hypotenuse.
5. Again some students will struggle with the opposite and adjacent. Earlier in the task the
opposite was the vertical side and now it is horizontal. Discuss how the position of the angle
alters the names of these two sides.
6. To this point students have not been introduced to the tan function on calculators. Remind
them that the tan ratio measures the ratio of the opposite side to the adjacent side and the
size of the side lengths is irrelevant. Guide them through finding the tan ratio for the angle
found using their calculators. Round off to 2 decimal places.
Remind students that the tan ratio is a measure of gradient. Why did those students
measuring from closer the structure have greater tan ratios?
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
7. Discuss which values in the equation tan  = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 have now been found. Solve for the
adjacent side. Once again some students will need this step scaffolded. Even students
familiar with solving equations of the form 6 =
18
𝑥
will be confused by the use of decimals.
Reflection 3
This task has introduced a number of concepts around the tangent ratio:
1. Similar right angled triangles have the same tan ratio for the corresponding angles
2. The tan ratio is not affected by the length of a side, or the size of a triangle, but by the
relative length of the opposite side compared to the adjacent side
3. When changing a right angled triangle, by modifying and fixing some side lengths and angles,
the other side lengths and angles, and the tan ratio, will be altered in various ways
4. The tan ratio allows the measurement of otherwise immeasurable lengths
Students record their learning as a journal entry. Encourage them to use correct mathematical
terms.