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:

3cos + 4sin cos( )R

Using the addition formula for cos(A – B) gives:

3cos + 4sin cos cos + sin sinR R

Equating the coefficients of cos θ and sin θ :

3 = cosR 3cos =R

4 = sinR 4sin =R

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

2 2= 3 + 4 = 5R

1 43= tan

So, using these values:

3 cos θ + 4 sin θ = 5 cos (θ – 53.1°)

On w/b1. Express 2cos θ + sin θ in the form R cos (θ – α).

Equations of the form a cos θ + b sin θ = ca) 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) 12sin 5cos sin( )R Using the addition formula for sin(θ – α) gives:

12sin 5cos sin cos cos sinR R

Equating the coefficients of cos θ and sin θ :

5 = sinR 5sin =R

12 = cosR 12cos =R

Using the following right-angled triangle:

α

R2 2= 5 +12 =13R

1 512= tan

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

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

13sin( 22.6 ) = 8 8

13sin( 22.6 ) =

(Using a calculator set to degrees:)

So

θ = (to 3 s.f.)

138sin6.22 1

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 - 22.6)º =

This happens when sin(Ө - 22.6)º =

so (Ө - 22.6)º =

Ө =

c) State the max value of 13 sin (θ – 22.6°) and find θ for which this occurs.

On w/ba) 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 pairs2. 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?

RememberExtra practice homework – Ex 5B p65 all do qu 2,3,5Challenge qu - 10