# Trigonometry 1: Definitions and properties

**AS Level**

- Understand and use the definitions of sine, cosine and tangent for all arguments; the sine and cosine rules; the area of a triangle in the form
^{1}/_{2}absinC

- Understand and use the sine, cosine and tangent functions; their graphs, symmetries and periodicity

**A Level**

- Work with radian measure, including use for arc length and area of sector
- Understand and use the standard small angle approximations of sine, cosine and tangent sinθ≈θ, cosθ≈1, tanθ≈θ where θ is in radians
- Know and use exact values of sin and cos for 0, π/6, π/4. π/3, π/2 and π and multiples thereof, and exact values of tan for 0, π/6, π/4. π/3, and π and multiples thereof
- Understand and use the definitions of secant, cosecant and cotangent and of arcsin, arccos and arctan; their relationships to sine, cosine and tangent; understanding of their graphs; their ranges and domains

### Trigonometry

All resources in this mathscentre collection contain comprehensive notes, with clear descriptions, together with relevant diagrams and examples. Students wishing to review, and consolidate, their knowledge and understanding of sequences will find them useful. Each topic includes a selection of questions to be completed, for which answers are provided.

The **Trigonometrical Ratios** resource introduces sine, cosine and tangent through right-angles triangles. It includes common triangles and corresponding values for sin, cos and tan, as well as how to use a calculator when working with these ratios.

### Trigonometric Ratios of Any Angle

This exhaustive PowerPoint presentation covers a large range of topics, including:

- how positive and negative angles are measured
- the labels given to each of the four quadrants
- what a circular function is and what the graphs of y=sin x, y=cos x and y=tan x look like
- the values of θ in the range 0
^{o}to 360^{o}which satisfy equations such as sin θ = sin 50^{o}using both the use of a circle and using the graph.

### Trigonometry: X=cosθ,Y=sinθ

This interactive excel file from The Virtual Textbook is intended to be presented to students on an interactive whiteboard. It uses the unit circle to explore values of cosθ and sinθ for angles between 0° and 360°, including trigonometric ratios of the angles 30°, 45° and 60°.

### Trigonometry

This mathcentre collection includes resources that cover **cosecant, secant and cotangent**, r**adians**, **trigonometric ratios of an angle of any size **and more.

Each resource contains comprehensive notes, relevant examples and exercises for students wishing to consolidate their understanding of trigonometry.

### Exploring Trigonometrical Graphs A12

This matching activity from the DfE Standards Unit introduces the period and amplitude of a trigonometrical graph and provides the opportunity for students to practice sketching trigonometrical graphs, and recognise translations, stretches and reflections from their equations.

Before attempting the activity, students should be familiar with the basic trig graphs and simple transformations of functions. The resource comes with detailed lesson plan and should take at least 45 minutes.

### Trigonometry 2

**RISP**s (**Ri**ch **S**tarting **P**oints) are intended to enrich the learning of mathematics in A Level classrooms. The teachers' notes suggest the most appropriate use for each activity. This collection contains the resource **Radians and Degrees, **which sets the task of finding an angle whose sine value is the same whether measured in radians or degrees.

### Trigonometrical Functions

This Further Thinking Questions resource, from Susan Wall, contains twelve problems requiring students to explore the properties of the functions of sine, cosine and tangent.

The problems are designed to encourage rich discussion in the classroom, by asking students to explain which is the odd one out, match functions with their graph, devise questions which give a certain set of solutions or state whether statements are true sometimes, always or never.