Outcomes
A student:

 MA5.31WM
uses and interprets formal definitions and generalisations when explaining solutions and/or conjectures

 MA5.32WM
generalises mathematical ideas and techniques to analyse and solve problems efficiently

 MA5.33WM
uses deductive reasoning in presenting arguments and formal proofs

 MA5.315MG
applies Pythagoras’ theorem, trigonometric relationships, the sine rule, the cosine rule and the area rule to solve problems, including problems involving three dimensions
Content
 Students:
 Apply Pythagoras' theorem and trigonometry to solve threedimensional problems in rightangled triangles (ACMMG276)
 solve problems involving the lengths of the edges and diagonals of rectangular prisms and other threedimensional objects
 use a given diagram to solve problems involving rightangled triangles in three dimensions
 check the reasonableness of answers to trigonometry problems involving rightangled triangles in three dimensions (Problem Solving)
 draw diagrams and use them to solve word problems involving rightangled triangles in three dimensions, including using bearings and angles of elevation or depression, eg 'From a point, A, due south of a flagpole 100 metres tall on level ground, the angle of elevation of the top of the flagpole is 35°. The top of the same flagpole is observed with an angle of elevation 22° from a point, B, due east of the flagpole. What is the distance from A to B?'
 check the reasonableness of answers to trigonometry word problems in three dimensions (Problem Solving)
 Use the unit circle to define trigonometric functions, and graph them, with and without the use of digital technologies (ACMMG274)
 prove that the tangent ratio can be expressed as a ratio of the sine and cosine ratios, ie \( \tan\theta = \dfrac{\sin\theta}{\cos\theta} \)
 use the unit circle and digital technologies to investigate the sine, cosine and tangent ratios for (at least) \( 0^{\circ} \le x \le 360^{\circ} \) and sketch the results
 compare the features of trigonometric curves, including periodicity and symmetry (Communicating, Reasoning)
 describe how the value of each trigonometric ratio changes as the angle increases from 0° to 360° (Communicating)
 recognise that trigonometric functions can be used to model natural and physical phenomena, eg tides, the motion of a swinging pendulum (Reasoning)
 investigate graphs of the sine, cosine and tangent functions for angles of any magnitude, including negative angles

use the unit circle or graphs of trigonometric functions to establish and use the following relationships for obtuse angles, where \( 0^{\circ} \le A \le 90^{\circ}\colon \)
\( \begin{align} \sin A &= \sin(180^{\circ}  A) \\ \cos A &= \cos(180^{\circ}  A) \\ \tan A &= \tan(180^{\circ}  A) \end{align} \)
 recognise that if \( \sin A \ge 0 \), then there are two possible values for \(A\), given \( 0^{\circ} \le A \le 180^{\circ} \) (Reasoning)
 determine the angle of inclination, \(\theta\), of a line on the Cartesian plane by establishing and using the relationship \(m=\tan \theta\) where \(m\) is the gradient of the line
 Solve simple trigonometric equations (ACMMG275)
 determine and use the exact sine, cosine and tangent ratios for angles of 30°, 45° and 60°
 solve problems in rightangled triangles using the exact sine, cosine and tangent ratios for 30°, 45° and 60° (Problem Solving)

prove and use the relationships between the sine and cosine ratios of complementary angles in rightangled triangles
\( \begin{align} \sin A &= \cos(90^{\circ}  A) \\ \cos A &= \sin(90^{\circ}  A) \end{align} \)  determine the possible acute and/or obtuse angle(s), given a trigonometric ratio
 Establish the sine, cosine and area rules for any triangle and solve related problems (ACMMG273)

prove the sine rule:
In a given triangle \(\textit{ABC}\), the ratio of a side to the sine of the opposite angle is a constant \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)  use the sine rule to find unknown sides and angles of a triangle, including in problems where there are two possible solutions when finding an angle
 recognise that if given two sides and a nonincluded angle, then two triangles may result, leading to two solutions when the sine rule is applied (Reasoning)

prove the cosine rule:
For a given triangle \(\textit{ABC}\),
\( \begin{align} a^2 &= b^2 + c^2  2bc\cos A \\ \cos A &= \frac{b^2 + c^2  a^2}{2bc} \end{align} \)  use the cosine rule to find unknown sides and angles of a triangle

prove and use the area rule to find the area of a triangle:
For a given triangle \( \textit{ABC} \), \(\text{Area of triangle} = \frac{1}{2} ab \sin C \)  select and apply the appropriate rule to find unknowns in nonrightangled triangles
 explain what happens if the sine, cosine and area rules are applied in rightangled triangles (Communicating, Reasoning)
 solve a variety of practical problems that involve nonrightangled triangles, including problems where a diagram is not provided
 use appropriate trigonometric ratios and formulas to solve twodimensional problems that require the use of more than one triangle, where the diagram is provided and where a verbal description is given (Problem Solving)
Background Information
Pythagoras' theorem is applied here to rightangled triangles in threedimensional space.
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.
The graphs of the trigonometric functions mark the transition from understanding trigonometry as the study of lengths and angles in triangles (as the word 'trigonometry' implies) to the study of waves, as will be developed in the Stage 6 calculus courses. Waves are fundamental to a vast range of physical and practical phenomena, such as light waves and all other electromagnetic waves, and to periodic phenomena such as daily temperatures and fluctuating sales over the year. The major importance of trigonometry lies in the study of these waves. The sine, cosine and tangent functions are plotted for a full revolution and beyond, so that their wave nature becomes clear.
Students are not expected to reproduce proofs of the sine, cosine and area rules. The cosine rule is a generalisation of Pythagoras' theorem. The sine rule is linked to the circumcircle and to circle geometry.
Students should realise that when the triangle is rightangled, the cosine rule becomes Pythagoras' theorem, the area formula becomes the simple 'half base times perpendicular height' formula, and the sine rule becomes a simple application of the sine function in a rightangled triangle.
The formula \(m = \tan \theta\) is a formula for gradient \( (m) \) in the Cartesian plane.