Teacher Delivery Guide Pure Mathematics: Trigonometry Specification Ref. Learning outcomes Notes Notation Exclusions PURE MATHEMATICS: TRIGONOMETRY (1) Basic trigonometry * Know how to solve right-angled triangles using trigonometry. Trig. functions Mt1 Be able to use the definitions of sin θ, cos θ and tan θ for any angle. By reference to the unit circle, . t2 Know and use the graphs of sin θ, cos θ and tan θ for all values of θ, their symmetries and periodicities. Stretches, translations and reflections of these graphs. Combinations of these transformations. Period. * Know and be able to use the exact values of sin θ and cos θ for θ = 0, 30, 45 , 60 and 90 and the exact values of tan θ for θ = 0, 30, 45 and 60. Area of triangle; sine and cosine rules t3 Know and be able to use the fact that the area of a triangle is given by ½. t4 Know and be able to use the sine and cosine rules. Use of bearings may be required. Identities t5 Understand and be able to use . e.g. solve for . t6 Understand and be able to use the identity . e.g. solve for .
know and apply the formula . know and apply Pythagoras’ Theorem and can thus extend this to the trigonometric identities and should then understand why they are called the Pythagorean Identities. Links Across the A-Level Course Surds – the manipulation of surds is a basic skill that then allows the learner to use the values of sine, cosine and tangent in exact form, particularly when solving trigonometric equations. Quadratic Equations – being able to comfortably solve quadratic equations specifically those in a function of the unknown like , or is important when solving trigonometric equations. Curve Sketching – although the initial skills are largely applied to polynomial equations the techniques developed here can easily be adapted and applied to the graphs of trigonometric functions. Functions – The use of function notation, range and domain, and the relationship between and will assist in the graphing of the trigonometric functions and understanding their connection to each other. Graph Transformations – as with curve sketching the initial work is done with polynomial functions, however exactly the same skill set applies to the transformation of trigonometric graphs. Circles – There is a link to circles particularly in the creation of the Pythagorean Identities. That is why these functions are correctly named the Circular Trigonometric Functions. Parametric Equations – A number of parametric equations are given in terms of trigonometric functions and so the understanding of how trigonometric functions work and link together becomes fundamental to this. Differentiation – This requires the learner to be able to differentiate trigonometric functions from first principles. Integration - the various techniques of integration will require a greater degree of manipulation of the trigonometric functions. Vectors – There is a small and limited use of trigonometry in finding the direction of vectors. Resolving Forces – The Mechanics strand involves using trigonometry to resolve forces horizontally and vertically, or parallel and perpendicular to an inclined plane.
Malin Christersson’s Math Site Introduces the idea of combining known angle facts to determine others. Includes a couple of questions at the end of the notes. t2 Chris Smith A very entertaining video that uses the circle and spaghetti to create a sine curve. t2 Geogebra Cosine Identity link the corresponding values of on the graph t2 Geogebra Sine Identity link the corresponding values of on the trig graph t2 Geogebra Tangent Identity link the corresponding values of on the graph . t2 Geogebra Trigonometry area formula Shows how to find the area of a triangle including step by step working. t3 Malin Christersson’s Math Site Demonstration of area of triangle and the sine/cosine rules. t3 and t4 Geogebra Interactive exercise that allows different triangles to be investigated with step by step working. t4 Geogebra Interactive exercise that allows different triangles to be investigated with step by step working. t4 Geogebra Interactive exercise that allows different triangles to be investigated with step by step working. t4 Geogebra Interactive exercise that allows different triangles to be investigated with step by step working. t4 Geogebra Shows the step by step proof for the first of the Pythagorean Identities using geogebra t6 Geogebra Gives the exact values of the trig functions around the circle. t8 Undergound Mathematics This problem looks at how , and are related to each other. By working on the task, students can combine graph sketching, the unit circle, identities and solving equations. t8