In this unit students will explore the importance of triangles, particularly right angled triangles, in the real world. Students will use practical measuring skills and calculations to find a pattern linking the ratio of the sides of a triangle with the angles. Trigonometry is introduced through the use of scale diagrams.
Trigonometry can be traced as far back as ancient Egypt, and possibly Babylonia. The name comes from the Greek words for triangle (trigonon – three angles) and measure (metron). Hellenist mathematicians, around 500-300 BC, applied trigonometry to the location of stars and other celestial objects.
Therefore, the origins of trigonometry lay in practical measurement tasks of finding unknown sides and angles, using right angled triangles. Medieval Persian mathematicians developed trigonometry as a separate field of mathematics, and much later, in the late 1500’s, the trigonometric (or circular) functions were developed. Many real-life situations are modelled by trigonometric functions, including the change of sea levels with a changing tide, and changing day length as a result of seasonal change.
Trigonometry relies on the conservation of ratios between corresponding sides of similar right-angled triangles. Consider the case of two similar right-angled triangles. (3, 4, 5 - shown in the diagram below) and (6, 8, 10) are Pythagorean triples since 32 + 42 = 52 and 62 + 82 = 102. Therefore, the triangles are right-angled. The matching angles of both triangles are also equal (for example, the two angles marked). Less obvious is the proportional relationship between matching side lengths. The ratios 3/6, 4/8, and 5/10 are all 1/2, which gives the scale factor mapping the larger triangle onto the smaller. The reciprocal ratios (6/3, 8/4, 10/5) are 2 which is the scale factor mapping the smaller triangle onto the larger.
In fact, the matching side ratios are the same for any right-angled triangle that is similar to those two triangles. For example, the triangles (1½, 2, 2½) and (12, 16, 20) have the same side ratios. All four triangles also have the same matching angles. To formalise this idea the sides of any right-angled triangle are labelled with reference to one of the angles.
The opposite side is always ‘on the other side’ to the angle and the adjacent side is always ‘next to’ the angle. The hypotenuse is always the longest side.
The trigonometric ratios sine, cosine and tangent are the invariant side ratios for any right-angled triangle with the same angle they refer to.
For example, in the (3, 4, 5) triangle, the angle in the left diagram can be referred to as:
Let’s imagine the (3, 4, 5) triangle enlarged by a factor of 0.2 or . That means that each side of the new triangle is one-fifth the original.
The unit triangle is useful for two reasons:
To find θ, look at the angle with a sine of 0.6, a cosine of 0.8, and a tangent of 0.75. Any one of the three ratios will do. On a scientific calculator key in:
Sin-1(0.6) = 36.87◦ (2 dp) Cos-1(0.8) = 36.87° Tan-1(0.75) = 36.87°
This unit may take longer than one week. Introducing trigonometry through scale diagrams mimics historical development which may help students to appreciate the sophistication of using trigonometric ratios to find unknown lengths and angles.
The ideas in this unit are explored further in the complementary Level 5 unit Using trigonometry.
The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:
The contexts presented in this unit are relevant to the real-world. However, you might choose to introduce new, more relevant, contexts later on in the unit, once students have developed the necessary understandings.
Te reo Māori kupu such as tāroa (hypotenuse), pātata (adjacent), tauaro (opposite), pākoki (trigonometry), koki hāngai (right angle), koki (angle), ōwehenga (ratio), and taurahi (scale factor) could be introduced in this unit and used throughout other mathematical learning.
In this session the importance of right-angled triangles is discussed.
Slide two introduces the problem of finding the steepness of the great pyramid. You may want to have a model of a pyramid to show how the steepness for slant height is steeper than slope along an edge. If you make a pyramid from modelling clay you can cut it vertically along a diagonal of the square base to illustrate.
You may need to discuss the measure “royal cubit” which was the length from the Pharaoh’s elbow to his longest finger (40 – 45 cm).
Ask students to use a ruler and protractor to make a scale drawing of both profiles. They should get a reasonable estimate of the angles, 51.84° for the slant height angles and 41.99° for the edge angle. These are the measures of the original pyramid which are different today due to erosion.
Discuss the exercise.
Look for students to notice that the triangles are similar as the corresponding angles are the same. Equivalence of ratios between the side length within each triangle and between the triangles is far less obvious.
2 is two thirds of 3 and 186 is two thirds of 279.
186 equals 93 x 2 and 279 equals 93 x 3. Note that 93 is the scale factor.
You might ask if the within and between relationships also hold for the hypotenuse. This gives an opportunity to apply the Pythagorean Theorem. The hypotenuse of the stick triangle equals √2² + 3² = 3.61 and the corresponding side of the pyramid triangle equals √186² + 279² = 335.32.
93 x 3.61 = 335.73 so the between relationship holds (allowing for rounding).
Are the within side ratios, vertical side ÷ hypotenuse, and horizontal side ÷ hypotenuse, the same for both triangles?
3 ÷ 3.61 = 0.83 and 2 ÷ 3.61 = 0.55 (stick triangle) and 279 ÷ 335.73 = 0.83 and 186 ÷ 335.73 = 0.55 (pyramid triangle). Therefore, the within side ratios are equal for both triangles. The equality of ratio within and between similar triangles is the basis of trigonometry.
After students complete the exercise, discuss what is always the same about similar right-angled triangles. Look for students to summarise the same angles and same side ratios.
Last lesson we learned about Thales. Today we explore a contribution from Hipparchus, another ancient Greek, who lived over 300 years later than Thales. His great mathematical invention was the unit circle.
Slide Two shows an application.
Students may recognise that the scale factor that maps the unit triangle onto the tree triangle equals 20 ÷ 0.866 = 23.09 (2 dp). Multiplying the opposite side of the unit triangle by that scale factor gives the height of the tree, 23.09 x 0.5 = 11.55 cubits. This method may seem inefficient compared to using Tan (30°) = x/20, but the ratio tan (30°) is a pre-calculated ratio.
You may wish to carry out a practical task of creating values for the opposite and adjacent sides using a metre-long hypotenuse. Hipparchus may well have engaged in a similar task to calculate his tables, albeit with more precision.
Angle | Adjacent side (metres) | Opposite side (metres) |
30° | ||
45° | ||
60° | ||
75° |
Students might notice that the values are ‘swapped’ for 30 and 60 degrees. They might also see that the adjacent side values decrease as angle increases while the opposite side values increase.
Your calculator already has these ratios of adjacent side ÷ hypotenuse, and opposite side ÷ hypotenuse, stored as cosine and sine. Since the hypotenuse is one the ratios are the same as the values you measured, e.g. For 60°, adjacent ÷ hypotenuse = 0.5 ÷ 1 = 0.5, and opposite ÷ hypotenuse = 0.866 ÷ 1 = 0.866.
The tree is 18 metres tall. The hypotenuse might be found using the Pythagorean Theorem, h = √18² + 18² , or cosine, cos(45°) = 18/h, or sine, sin(45°) = 18/h.
This session explores the nine basic problem types that can be solved using cosine, sine and tangent ratios. The important question is:
How many measurements of a right-angled triangle are needed to draw the triangle?
For example, 37° might be thrown first. Students might draw this:
Some students may recognise that the angles are now set at 90°, 37°, and 63° as the sum of internal angles must equal 180°. However, without a side length the scale of the triangle is unknown.
Next 8cm is rolled. Students might draw three different figures.
Try three examples of trying to draw triangles progressively from measurement information. The important generalisations to make are that it is possible to draw an exact triangle if:
Knowing all three angles gives similar triangles but the size is unclear.
At this point students have not encountered tangent. Recap on what measures are given for sine and cosine to be relevant to solve a problem.
Sine is the ratio of opposite divided by hypotenuse. Cosine is the ratio of adjacent divided by hypotenuse. Show the relationships like this. After Sine and Cosine are represented by arrows ask: What ratio is missing?
Students might notice that they do not know a ratio for the relationship between opposite and adjacent sides. Tell your students that tangent is the name for the ratio of opposite divided by adjacent. You might introduce the mnemonic SOH-CAH-TOA as a helpful way to remember the three basic trigonometric ratios. Proceed to Slides 7-9 of PowerPoint Three so student see how the tangent ratio can be applied.
Students may need more practice with identifying the ratio required for a given situation and writing the equation to be solved.
At this point students need to solve linear equations like cos(56°) = x/16 and cos(56°) = 28/x to find unknown side lengths. Common issues students might face include:
Watch for these issues as your students work on the examples in PowerPoint Four. Pause the video where you please to allow for students to work on sub-tasks and to have discussion.
After going through the PowerPoint challenge your students to may up a similar problem for a partner to solve. Ask them to provide a full solution with all steps. Choose some examples to share with the class and put the others into a book of trigonometry problems.
For more algebra practice ask your students to work on the four learning objects in the Visual Linear Algebra suite (https://meaningfulmaths.nt.edu.au/mmws/nz/content/visual-linear-algebra). Familiarity with solving linear equations is important for other aspects of the Level 5 curriculum, such as reasoning with rates and ratios, finding areas and perimeters, and conversions in measurement, and interpreting scatterplots in statistics.
Dear families and whānau,
Recently we have been using practical measuring skills and calculations to find a pattern linking the ratio of the sides of a triangle with the angles. We have used scale diagrams to learn about trigonometry. Ask your child to share their learning with you.
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/introducing-trigonometry at 8:43pm on the 26th February 2024