1. Trigonometry: Addition formula for Rcos (x+a) and R(x+a)
KUS objectives
BAT write expressions of the form acosθ + bsinθ, where a and b are constants, as a sine or cosine function only
BAT solve equations of the form acosθ + bsinθ, or find their max/min value
Starter:

2. R sin (𝜃+ ∝) =𝑹 sin 𝜃 𝐜𝐨𝐬 ∝ +𝑹 cos 𝜃 𝐬𝐢𝐧 ∝
WB1 Show that 3 sin 𝜃 +4 cos 𝜃 can be expressed in the form 𝑅 sin 𝜃+∝
where 𝑅>0, 0< 𝛼< 𝜋 2
R sin (𝜃+ ∝) =𝑹 sin 𝜃 𝐜𝐨𝐬 ∝ +𝑹 cos 𝜃 𝐬𝐢𝐧 ∝
Compare each term – they must be equal!
𝟑 𝑠𝑖𝑛𝜃 − 𝟒 𝑐𝑜𝑠𝜃 α 𝑅 3 4
So in the triangle, the Hypotenuse is R…
𝑹 𝒔𝒊𝒏 ∝ =𝟔
𝑹 𝐜𝐨𝐬 ∝ =𝟑
→ 𝑡𝑎𝑛 ∝ = 6 3 =2
𝑅= = 5
→ ∝ =1.107
SO 𝟑 𝑠𝑖𝑛𝜃 − 𝟒 𝑐𝑜𝑠𝜃 =5 sin 𝜃+1.107

3. R cos (𝜃− ∝) =𝑹 cos 𝜃 𝐜𝐨𝐬 ∝ +𝑹 sin 𝜃 𝐬𝐢𝐧 ∝
WB2:
a) Express 2 cos 𝜃 +5 sin 𝜃 can be expressed in the form 𝑅 cos 𝜃−∝
where 𝑅>0, 0< 𝛼< 𝜋 2
b) Hence, solve 2 cos 𝜃 +5 sin 𝜃 =3 in the range 0<𝜃<2π
R cos (𝜃− ∝) =𝑹 cos 𝜃 𝐜𝐨𝐬 ∝ +𝑹 sin 𝜃 𝐬𝐢𝐧 ∝
Compare each term carefully- get sin and cos in correct order
𝟐 cos 𝜃 𝟓 sin 𝜃 α 𝑅 2 5
So in the triangle, the Hypotenuse is R…
𝑹 𝒔𝒊𝒏 ∝ =𝟓
𝑹 𝐜𝐨𝐬 ∝ =𝟐
→ 𝑡𝑎𝑛 ∝ = 5 2
𝑅= = 29
→ ∝ =1.190
SO 𝟐 cos 𝜃 +𝟓 sin 𝜃 = sin 𝜃−1.190
cos 𝜃−1.190 =
𝜃− = 0.980, 5.303
𝜃 = 2.17, 6.49

4. Compare each term – they must be equal!
WB 3a
a) Show that you can express sin 𝜃 − 3 cos 𝜃 in the form 𝑅 sin 𝜃+∝
where 𝑅>0, 0< 𝛼<90
b) Hence, sketch the graph of y= sin 𝑥 − 3 cos 𝑥
R sin (𝜃+ ∝) =𝑹 sin 𝜃 𝐜𝐨𝐬 ∝ +𝑹 cos 𝜃 𝐬𝐢𝐧 ∝
Compare each term – they must be equal!
𝟏 sin 𝜃 − 𝟑 𝑐𝑜𝑠𝜃
→ 𝑡𝑎𝑛 ∝ = 𝟑 1 = 𝟑
𝑅= = 2
𝑹 𝒔𝒊𝒏 ∝ = 𝟑
𝑹 𝐜𝐨𝐬 ∝ =𝟏
→ ∝ = 𝜋 3
SO sin 𝜃 − 𝟑 𝑐𝑜𝑠𝜃 =2 sin 𝜃+ 𝜋 3

5. WB3b: b) Sketch the graph of: 𝑠𝑖𝑛𝑥− 3 𝑐𝑜𝑠𝑥 2sin⁡ 𝑥− 𝜋 3
𝑠𝑖𝑛𝑥− 3 𝑐𝑜𝑠𝑥
2sin⁡ 𝑥− 𝜋 3
= Sketch the graph of: 1
y=sin⁡𝑥
Start out with sinx -1
π/2 π
3π/2 2π
y=sin⁡ 𝑥− 𝜋 3 1
Translate π/3 units right -1
π/3
π/2 π
4π/3
3π/2 2π 2
y=2sin⁡ 𝑥− 𝜋 3
Vertical stretch, scale factor 2 1 -1
π/3
π/2 π
4π/3
3π/2 2π -2
At the y-intercept, x = 0
2sin⁡ − 𝜋 3
=− 3

6. Remember to get the + or – signs the correct way round!
Which one should be used?
𝑎𝑠𝑖𝑛𝜃±𝑏𝑐𝑜𝑠𝜃
𝑅𝑠𝑖𝑛 𝜃±𝛼
𝑎𝑐𝑜𝑠𝜃±𝑏𝑠𝑖𝑛𝜃
𝑅𝑐𝑜𝑠 𝜃∓𝛼
Whichever ratio is at the start, change the expression into a function of that (This makes solving problems easier)
Remember to get the + or – signs the correct way round!

7. Which addition formula should we use?
WB4: Solve the equation 9𝑠𝑖𝑛𝜃 −6𝑐𝑜𝑠𝜃=7 𝑓𝑜𝑟 0<𝜃<2π
Which addition formula should we use?
R sin (𝜃− ∝) =𝑹 sin 𝜃 𝐜𝐨𝐬 ∝ −𝑹 cos 𝜃 𝐬𝐢𝐧 ∝
𝟗 𝑠𝑖𝑛𝜃 − 𝟔 𝑐𝑜𝑠𝜃 =7
𝑹 𝒔𝒊𝒏 ∝ =𝟔
𝑹 𝐜𝐨𝐬 ∝ =𝟗
→ 𝑡𝑎𝑛 ∝ = 6 9
𝑅= =3 13
→ ∝ =0.588
So we have 𝑠𝑖𝑛 𝜃−5.88 =7
→ 𝑠𝑖𝑛 𝜃−0.588 =
→ 𝜃− =0.704, 2.438
→ 𝜃 =1.292, 3.026

8. Which addition formula should we use?
WB5: Find the maximum value of the value of the expression 2sinx + 3cosx, and find the value of x (0<x<2π) at which the maximum occurs
Which addition formula should we use?
R sin (𝑥+ ∝) =𝑹 sin 𝑥 𝐜𝐨𝐬 ∝ + 𝑹 cos 𝑥 𝐬𝐢𝐧 ∝
𝟐 sin 𝑥 𝟑 cos 𝑥
→ 𝑡𝑎𝑛 ∝ = 3 2
𝑹 𝒔𝒊𝒏 ∝ =𝟑
𝑹 𝐜𝐨𝐬 ∝ =𝟐
𝑅= = 13
→ ∝ =0.983
SO 𝟐 sin 𝑥 + 𝟑 cos 𝑥 = sin 𝜃+0.983
This has a maximum when 𝜃+0.983= 𝜋 2 → 𝜃= in the range 0<𝑥<2𝜋
→ cos 𝜃 = 13

9. R cos (𝜃− ∝) =𝑹 cos 𝜃 𝐜𝐨𝐬 ∝ −𝑹 sin 𝜃 𝐬𝐢𝐧 ∝
WB6:
a) Express 12 cos 𝜃 +5 sin 𝜃 in the form 𝑅 cos 𝜃−∝
where 𝑅>0, 0< 𝛼<90
b) Hence, find the maximum value of 12 cos 𝜃 +5 sin 𝜃 and the smallest positive
value of  at which it arises
R cos (𝜃− ∝) =𝑹 cos 𝜃 𝐜𝐨𝐬 ∝ −𝑹 sin 𝜃 𝐬𝐢𝐧 ∝
𝟏𝟐 cos 𝜃 𝟓 sin 𝜃
→ 𝑡𝑎𝑛 ∝ = 5 12
𝑹 𝒔𝒊𝒏 ∝ =𝟓
𝑹 𝐜𝐨𝐬 ∝ =𝟏𝟐
𝑅= =13
→ ∝ =0.395
SO 𝟏𝟐 cos 𝜃 + 𝟓 sin 𝜃 =13 cos 𝜃−0.935
This has a maximum when 𝜃=0.935
→ 13 cos 𝜃− =13

10. Which addition formula should we use?
WB7: Express 4𝑐𝑜𝑠𝑥+3𝑠𝑖𝑛𝑥 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑜𝑟𝑚 𝑟𝑐𝑜𝑠(𝑥−,) 𝑤ℎ𝑒𝑟𝑒 𝑅>0 𝑎𝑛𝑑 0<∝< 𝜋 2 .
Hence solve 4 𝑐𝑜𝑠𝑥+3𝑠𝑖𝑛𝑥=2 𝑓𝑜𝑟 0<𝑥<𝜋, giving your answers to 2.d.p
Which addition formula should we use?
R cos (𝑥−∝) =𝑹 cos 𝑥 𝐜𝐨𝐬 ∝ + 𝑹 sin 𝑥 𝐬𝐢𝐧 ∝
𝟒 cos 𝑥 𝟑 sin 𝑥
→ 𝑡𝑎𝑛 ∝ = 3 4
𝑹 𝒔𝒊𝒏 ∝ =𝟑
𝑹 𝐜𝐨𝐬 ∝ =𝟒
𝑅= =5
→ ∝ =0.643
SO 𝟒 cos 𝑥 + 𝟑 sin 𝑥 =5 cos 𝑥− =2
→cos 𝑥− = 2 5
→ 𝑥− =1.159, 5.124, ….
→𝑥 =1.80, only in the given range

11. self-assess using: R / A / G ‘I am now able to ____ .
KUS objectives
BAT write expressions of the form acosθ + bsinθ, where a and b are constants, as a sine or cosine function only
BAT solve equations of the form acosθ + bsinθ, or find their max/min value
self-assess using: R / A / G
‘I am now able to ____ .
To improve I need to be able to ____’