Outcomes
A student:

 MA5.11WM
uses appropriate terminology, diagrams and symbols in mathematical contexts

 MA5.12WM
selects and uses appropriate strategies to solve problems

 MA5.13WM
provides reasoning to support conclusions that are appropriate to the context

 MA5.110MG
applies trigonometry, given diagrams, to solve problems, including problems involving angles of elevation and depression
Content
 Students:
 Use similarity to investigate the constancy of the sine, cosine and tangent ratios for a given angle in rightangled triangles (ACMMG223)
 identify the hypotenuse, adjacent sides and opposite sides with respect to a given angle in a rightangled triangle in any orientation
 label sides of rightangled triangles in different orientations in relation to a given angle (Communicating)
 label the side lengths of a rightangled triangle in relation to a given angle, eg side \(c\) is opposite angle \(C\)
 define the sine, cosine and tangent ratios for angles in rightangled triangles
 use similar triangles to investigate the constancy of the sine, cosine and tangent ratios for a given angle in rightangled triangles
 use trigonometric notation, eg \(\sin{C}\)
 use a calculator to find approximations of the trigonometric ratios for a given angle measured in degrees
 use a calculator to find an angle correct to the nearest degree, given one of the trigonometric ratios for the angle
 Apply trigonometry to solve rightangled triangle problems (ACMMG224)
 select and use appropriate trigonometric ratios in rightangled triangles to find unknown sides, including the hypotenuse
 select and use appropriate trigonometric ratios in rightangled triangles to find unknown angles correct to the nearest degree
 Solve rightangled triangle problems, including those involving angles of elevation and depression (ACMMG245)
 identify angles of elevation and depression
 interpret diagrams in questions involving angles of elevation and depression (Reasoning)
 connect the alternate angles formed when parallel lines are cut by a transversal with angles of elevation and depression (Reasoning)
 solve a variety of practical problems, including those involving angles of elevation and depression, when given a diagram
Background Information
The definitions of the trigonometric ratios rely on the angle test for similarity, and trigonometry is, in effect, automated calculations with similarity ratios. The content is thus strongly linked with ratio and with scale drawing.
The fact that the other angles and sides of a rightangled triangle are completely determined by giving two other measurements is justified by the four standard congruence tests.
Trigonometry is introduced through similar triangles, with students calculating the ratio of two sides and realising that this remains constant for a given angle.
Purpose/Relevance of Substrand
Trigonometry allows the user to determine unknown sides and angles in both rightangled and nonrightangled triangles, and so solve related twodimensional and threedimensional realworld problems. It played a key role in the development of measurement in astronomy and land surveying and, with the trigonometric functions, is very important in parts of pure mathematics and applied mathematics. These are, in turn, very important to many branches of science and technology. Trigonometry or trigonometric functions are used in a broad range of areas, including astronomy, navigation (on ships, on aircraft and in space), the analysis of financial markets, electronics, statistics, biology, medicine, chemistry, meteorology, many physical sciences, architecture, economics, and different branches of engineering.
Language
In Stage 5, students are expected to know and use the sine, cosine and tangent ratios. The reciprocal ratios, cosecant, secant and cotangent, are introduced in selected courses in Stage 6.
Emphasis should be placed on correct pronunciation of 'sin' as 'sine'.
Initially, students should write the ratio of sides for each of the trigonometric ratios in words, eg \( \tan\theta = \dfrac{ \textrm{side opposite angle}~\theta}{\textrm{side adjacent to angle}~\theta} \). Abbreviations can be used once students are more familiar with the trigonometric ratios.
When expressing fractions in English, the numerator is said first, followed by the denominator. However, in many Asian languages (eg Chinese, Japanese), the opposite is the case: the denominator is said before the numerator. This may lead to students from such language backgrounds mistakenly using the reciprocal of the intended trigonometric ratio.
Students should be explicitly taught the meaning of the phrases 'angle of elevation' and 'angle of depression'. While the meaning of 'angle of elevation' may be obvious to many students, the meaning of 'angle of depression' as the angle through which a person moves (depresses) their eyes from the horizontal line of sight to look downwards at the required point may not be as obvious to some students.
Teachers should explicitly demonstrate how to deconstruct the large descriptive noun groups frequently associated with angles of elevation and depression in word problems, eg 'The angle of depression of a ship 200 metres out to sea from the top of a cliff is 25°'.
Students may find some of the terminology encountered in word problems involving trigonometry difficult to interpret, eg 'base/foot of the mountain', 'directly overhead', 'pitch of a roof', 'inclination of a ladder'. Teachers should provide students with a variety of word problems and they should explain such terms explicitly.
The word 'trigonometry' is derived from two Greek words meaning 'triangle' and 'measurement'.
The word 'cosine' is derived from the Latin words complementi sinus, meaning 'complement of sine', so that \(\cos{40^\circ}=\sin{50^\circ}\).