In this unit, students will explore the use of trigonometry to find unknown sides and angles in right-angled triangles.
Maths skills required from other strands:
This unit applies the concepts developed in the Level 5 unit Introducing Trigonometry.
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, such as sea levels as the tide changes, and day length as seasons 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) 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 by division, 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, referring to as the angle in the left diagram:
Let’s imagine the (3, 4, 5) triangle enlarged by a factor of 0.2 or 2/5. 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 up 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 learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to support students include:
Te reo Māori kupu such as koki hāngai (right angle), ōwehenga (ratio), and pākoki (trigonometry) could be introduced in this unit and used throughout other mathematical learning.
Give students, working in pairs, a set of matching activities (Copymaster Two). There are seven matching sets that might be used as the main part of a lesson or progressively over several days. Students need to shuffle the cards then attempt to:
Alternatively, you can delete parts of the sheet and the students can be challenged to:
If you choose to introduce Copymaster Three as a set of smaller tasks to be explored over a few days, consider using the time to also address specific, demonstrated learning needs with small groups of students (perhaps those you identified in session 1).
Practical problem-solving activities
The students use their trigonometric ability to solve practical problems:
Dear families and whānau,
Recently we have been exploring how to use trigonometry to find unknown sides and angles in right-angled triangles. Ask your child to share their learning with you.
Printed from https://nzmaths.co.nz/resource/using-trigonometry at 5:03am on the 25th April 2024