Circle the trig identities that you will need to learn for the Core 3 & 4 Exams. Remember some are given in the formula booklet so no need to learn.

Aims: To be able to express asinθ +bcosθ in the form Rsin(θ±α) and Rcos(θ±α). To be able to solve trig equations in the forms Rsin(θ±α) and Rcos(θ±α). To be able to state the max and min values of trig equations in the forms Rsin(θ±α) and Rcos(θ±α). Trigonometry Lesson 5 12 sin θ – 5 cos θ = 8

Expressions of the form a cos θ + b sin θ We are unable to solve functions of the form a cos θ + b sin θ so we need to change them into an expression containing just a cos or sin. Hence we express asinθ + bcosθ in the form Rsin(θ±α) or Rcos(θ±α) whereby R and α are constants to be found. To find the values of R and α we use the compound identities, a bit of Pythagoras and some basic trigonometry. Recall the Compound Identities, these will come in useful!

Equations of the form a cos θ + b sin θ = c Start by writing this as an identity: Using the addition formula for cos( A – B ) gives: Equating the coefficients of cos θ and sin θ : Express 3 cos θ + 4 sin θ in the form R cos ( θ – α ).

Equations of the form a cos θ + b sin θ = c Using these in a right-angled triangle gives: α R So, using these values: 3 cos θ + 4 sin θ = 5 cos ( θ – 53.1°)

On w/b 1. Express 2cos θ + sin θ in the form R cos ( θ – α ).

Equations of the form a cos θ + b sin θ = c a)Express 12 sin θ – 5 cos θ in the form R sin( θ – α ). b)Solve the equation 12 sin θ – 5 cos θ = 8 in the interval 0 < θ < 360°. c)State the max value and find θ for which this occurs. a) Using the addition formula for sin( θ – α ) gives: Equating the coefficients of cos θ and sin θ :

Using the following right-angled triangle: α R So, using these values 12 sin θ – 5 cos θ = 13 sin ( θ – 22.6°) Equations of the form a cos θ + b sin θ = c

b) Using the form found in part a) we can write the equation 12 sin θ – 5 cos θ = 8 as (Using a calculator set to degrees:) So θ = (to 3 s.f.) y=8/13 b) Solve the equation 12 sin θ – 5 cos θ = 8 in the interval 0 < θ < 360°

Equations of the form a cos θ + b sin θ = c c) The maximum value occurs when sin of the angle is _______ So 13sin( Ө ) º = This happens when sin(Ө )º = so (Ө )º = Ө = c) State the max value of 13 sin ( θ – 22.6°) and find θ for which this occurs.

On w/b a)Express 5cos θ + 6 sin θ in the form R sin( θ + α ). Where α is acute. b)State the max value and find θ for which this occurs. 1.Complete wheel puzzle in pairs 2.Treasure hunt in 3 or 4’s. 3.Extra practice – Ex 5B p65

To solve expressions of the form a cos θ + b sin θ write it first as: R cos ( θ – α ) or R cos ( θ + α ) or R sin ( θ – α ) or R sin ( θ + α ) You will be told which one to use in the exam question Have you understood today’s lesson? Remember Extra practice homework – Ex 5B p65 all do qu 2,3,5 Challenge qu - 10