R Addition Formulae

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R Addition Formulae

We use the R formulae when we have a set of sin⁡textcolor{blue}{sin} and cos⁡textcolor{limegreen}{cos} functions which we want to collect as a single trig function. Essentially, this is the Addition and Double Angle Formulae section, in reverse.

Make sure you are happy with the following topics before continuing.

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Origins of the R Formulae

Don’t worry, we’re not teaching you another equation – we’re teaching you a new trick for an old equation.

In the Addition and Double Angle Formulae section, we claimed that we could split sin⁡(A±B)textcolor{blue}{sin} (textcolor{purple}{A} pm textcolor{orange}{B}) and cos⁡(A±B)textcolor{limegreen}{cos}(textcolor{purple}{A} pm textcolor{orange}{B}) into separate sin⁡A,sin⁡B,cos⁡Atextcolor{blue}{sin} textcolor{purple}{A}, textcolor{blue}{sin} textcolor{orange}{B}, textcolor{limegreen}{cos} textcolor{purple}{A} and cos⁡Btextcolor{limegreen}{cos} textcolor{orange}{B} terms.

We’re now going to look at how to collect a linear combination of those terms into one expression, Rsin⁡(θ±α)textcolor{maroon}{R}textcolor{blue}{sin} (textcolor{purple}{theta} pm textcolor{orange}{alpha}) or Rcos⁡(θ∓α)textcolor{maroon}{R}textcolor{limegreen}{cos} (textcolor{purple}{theta} mp textcolor{orange}{alpha}).

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Using R Formulae

So, let’s say we have a general expression,

asin⁡θ±bcos⁡θatextcolor{blue}{sin} textcolor{purple}{theta} pm btextcolor{limegreen}{cos} textcolor{purple}{theta}

Surely we can express that as

Rcos⁡αsin⁡θ±Rsin⁡αcos⁡θtextcolor{maroon}{R}textcolor{limegreen}{cos} textcolor{orange}{alpha} textcolor{blue}{sin} textcolor{purple}{theta} pm textcolor{maroon}{R}textcolor{blue}{sin} textcolor{orange}{alpha} textcolor{limegreen}{cos} textcolor{purple}{theta}

where a=Rcos⁡αa = textcolor{maroon}{R}textcolor{limegreen}{cos} textcolor{orange}{alpha} and b=Rsin⁡αb = textcolor{maroon}{R}textcolor{blue}{sin} textcolor{orange}{alpha}, right?

Rcos⁡αsin⁡θ±Rsin⁡αcos⁡θ=R(cos⁡αsin⁡θ±sin⁡αcos⁡θ)=Rsin⁡(θ±α)textcolor{maroon}{R}textcolor{limegreen}{cos} textcolor{orange}{alpha} textcolor{blue}{sin} textcolor{purple}{theta} pm textcolor{maroon}{R}textcolor{blue}{sin} textcolor{orange}{alpha} textcolor{limegreen}{cos} textcolor{purple}{theta} = textcolor{maroon}{R}(textcolor{limegreen}{cos} textcolor{orange}{alpha} textcolor{blue}{sin} textcolor{purple}{theta} pm textcolor{blue}{sin} textcolor{orange}{alpha} textcolor{limegreen}{cos} textcolor{purple}{theta}) = textcolor{maroon}{R}textcolor{blue}{sin} (textcolor{purple}{theta} pm textcolor{orange}{alpha})

Then, we can find a value of αtextcolor{orange}{alpha}, or rather, tan⁡αtextcolor{red}{tan} textcolor{orange}{alpha}, by finding ba=Rsin⁡αRcos⁡α=tan⁡αdfrac{b}{a} = dfrac{textcolor{maroon}{R}textcolor{blue}{sin} textcolor{orange}{alpha}}{textcolor{maroon}{R}textcolor{limegreen}{cos} textcolor{orange}{alpha}} = textcolor{red}{tan} textcolor{orange}{alpha}, so α=tan⁡−1batextcolor{orange}{alpha} = tan ^{-1}dfrac{b}{a}.

We can also find a value of Rtextcolor{maroon}{R}, by doing the calculation a2+b2=R2cos⁡2α+R2sin⁡2α=R2(cos⁡2α+sin⁡2α)=R2a^2 + b^2 = textcolor{maroon}{R}^2textcolor{limegreen}{cos} ^2 textcolor{orange}{alpha} + textcolor{maroon}{R}^2textcolor{blue}{sin} ^2 textcolor{orange}{alpha} = textcolor{maroon}{R}^2(textcolor{limegreen}{cos} ^2 textcolor{orange}{alpha} + textcolor{blue}{sin} ^2 textcolor{orange}{alpha}) = textcolor{maroon}{R}^2, so R=a2+b2textcolor{maroon}{R} = sqrt{a^2 + b^2}.

Substituting our values of αtextcolor{orange}{alpha} and Rtextcolor{maroon}{R} gives us an expression for asin⁡θ±bcos⁡θatextcolor{blue}{sin} textcolor{purple}{theta} pm btextcolor{limegreen}{cos} textcolor{purple}{theta} in the form Rsin⁡(θ±α)textcolor{maroon}{R}textcolor{blue}{sin} (textcolor{purple}{theta} pm textcolor{orange}{alpha}).

 

We can do a similar process to find an expression in Rcos⁡(θ∓α)color{maroon}Rcolor{limegreen}cos color{grey}(color{purple}theta color{grey}mp color{orange}alphacolor{grey}), also.

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Factor Formulae

We’ve also got four new formulae…

sin⁡A+sin⁡B=2sin⁡(A+B2)cos⁡(A−B2)textcolor{blue}{sin} textcolor{purple}{A} + textcolor{blue}{sin} textcolor{orange}{B} = 2textcolor{blue}{sin} left( dfrac{textcolor{purple}{A} + textcolor{orange}{B}}{2}right) textcolor{limegreen}{cos} left( dfrac{textcolor{purple}{A} – textcolor{orange}{B}}{2}right)

sin⁡A−sin⁡B=2cos⁡(A+B2)sin⁡(A−B2)textcolor{blue}{sin} textcolor{purple}{A} – textcolor{blue}{sin} textcolor{orange}{B} = 2textcolor{limegreen}{cos} left( dfrac{textcolor{purple}{A} + textcolor{orange}{B}}{2}right) textcolor{blue}{sin} left( dfrac{textcolor{purple}{A} – textcolor{orange}{B}}{2}right)

cos⁡A+cos⁡B=2cos⁡(A+B2)cos⁡(A−B2)textcolor{limegreen}{cos} textcolor{purple}{A} + textcolor{limegreen}{cos} textcolor{orange}{B} = 2textcolor{limegreen}{cos} left( dfrac{textcolor{purple}{A} + textcolor{orange}{B}}{2}right) textcolor{limegreen}{cos} left( dfrac{textcolor{purple}{A} – textcolor{orange}{B}}{2}right)

cos⁡A−cos⁡B=2sin⁡(A+B2)sin⁡(A−B2)textcolor{limegreen}{cos} textcolor{purple}{A} – textcolor{limegreen}{cos} textcolor{orange}{B} = 2textcolor{blue}{sin} left( dfrac{textcolor{purple}{A} + textcolor{orange}{B}}{2}right) textcolor{blue}{sin} left( dfrac{textcolor{purple}{A} – textcolor{orange}{B}}{2}right)

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Example 1: Converting With R Formulae

That might’ve been a lot of information at once, so let’s try an example.

Express 8sin⁡x+6cos⁡x8textcolor{blue}{sin} x + 6textcolor{limegreen}{cos} x in the form Rsin⁡(x+α)textcolor{maroon}{R}textcolor{blue}{sin} (x + textcolor{orange}{alpha}), where 0°≤α≤90°0° leq textcolor{orange}{alpha} leq 90°.

[4 marks]

Let Rcos⁡α=8textcolor{maroon}{R}textcolor{limegreen}{cos} textcolor{orange}{alpha} = 8, and Rsin⁡α=6textcolor{maroon}{R}textcolor{blue}{sin} textcolor{orange}{alpha} = 6.

So, then, R2=R2(cos⁡2α+sin⁡2α)=R2cos⁡2α+R2sin⁡2α=82+62=100textcolor{maroon}{R}^2 = textcolor{maroon}{R}^2(textcolor{limegreen}{cos} ^2 textcolor{orange}{alpha} + textcolor{blue}{sin} ^2 textcolor{orange}{alpha}) = textcolor{maroon}{R}^2textcolor{limegreen}{cos} ^2 textcolor{orange}{alpha} + textcolor{maroon}{R}^2textcolor{blue}{sin} ^2 textcolor{orange}{alpha} = 8^2 + 6^2 = 100, giving R=10textcolor{maroon}{R} = 10.

Also, α=tan⁡−1(tan⁡α)=tan⁡−1(Rsin⁡αRcos⁡α)=tan⁡−168=36.87°textcolor{orange}{alpha} = tan ^{-1}(textcolor{red}{tan} textcolor{orange}{alpha}) = tan ^{-1} left( dfrac{textcolor{maroon}{R}textcolor{blue}{sin} textcolor{orange}{alpha}}{textcolor{maroon}{R}textcolor{limegreen}{cos} textcolor{orange}{alpha}}right) = tan ^{-1} dfrac{6}{8} = 36.87°.

 

So, to summarise, we have

8sin⁡x+6cos⁡x=10sin⁡(x+36.87°)8textcolor{blue}{sin} x + 6textcolor{limegreen}{cos} x = 10textcolor{blue}{sin} (x + 36.87°)

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Example 2: Converting With R Formulae (Again)

Express 8sin⁡x+6cos⁡x8textcolor{blue}{sin} x + 6textcolor{limegreen}{cos} x in the form Rcos⁡(x−α)textcolor{maroon}{R}textcolor{limegreen}{cos} (x – textcolor{orange}{alpha}), where 0°≤α≤90°0° leq textcolor{orange}{alpha} leq 90°.

[4 marks]

Let Rsin⁡α=8textcolor{maroon}{R}textcolor{blue}{sin} textcolor{orange}{alpha} = 8, and Rcos⁡α=6textcolor{maroon}{R}textcolor{limegreen}{cos} textcolor{orange}{alpha} = 6.

So, then, R2=R2(cos⁡2α+sin⁡2α)=R2cos⁡2α+R2sin⁡2α=62+82=100textcolor{maroon}{R}^2 = textcolor{maroon}{R}^2(textcolor{limegreen}{cos} ^2 textcolor{orange}{alpha} + textcolor{blue}{sin} ^2 textcolor{orange}{alpha}) = textcolor{maroon}{R}^2textcolor{limegreen}{cos} ^2 textcolor{orange}{alpha} + textcolor{maroon}{R}^2textcolor{blue}{sin} ^2 textcolor{orange}{alpha} = 6^2 + 8^2 = 100, giving R=10textcolor{maroon}{R} = 10.

Also, α=tan⁡−1(tan⁡α)=tan⁡−1(Rsin⁡αRcos⁡α)=tan⁡−186=53.13°textcolor{orange}{alpha} = tan ^{-1}(textcolor{red}{tan} textcolor{orange}{alpha}) = tan ^{-1} left( dfrac{textcolor{maroon}{R}textcolor{blue}{sin} textcolor{orange}{alpha}}{textcolor{maroon}{R}textcolor{limegreen}{cos} textcolor{orange}{alpha}}right) = tan ^{-1} dfrac{8}{6} = 53.13°.

 

So, to summarise, we have 8sin⁡x+6cos⁡x=10cos⁡(x−53.13°)8textcolor{blue}{sin} x + 6textcolor{limegreen}{cos} x = 10textcolor{limegreen}{cos} (x – 53.13°).

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R Addition Formulae Example Questions

Question 1: Convert 5cos⁡x−3sin⁡x5cos x – 3sin x into the form Rcos⁡(x+α)Rcos (x + alpha).

[4 marks]

A Level AQAEdexcelOCR

Let a=5=Rcos⁡αa = 5 = Rcos alpha and b=3=Rsin⁡αb = 3 = Rsin alpha.

Then,

R=52+32=34R = sqrt{5^2 + 3^2} = sqrt{34} and α=tan⁡−135=30.96°alpha = tan ^{-1} dfrac{3}{5} = 30.96°

 

Giving

5cos⁡x−3sin⁡x=34cos⁡(x+30.96°)5cos x – 3sin x = sqrt{34}cos (x + 30.96°)

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Question 2: Using factor formulae, convert 6sin⁡(π4)sin⁡(3π2)6sin left( dfrac{pi}{4}right) sin left( dfrac{3pi}{2} right) in surd form.

[3 marks]

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6sin⁡(π4)sin⁡(3π2)6sin left( dfrac{pi}{4}right) sin left( dfrac{3pi}{2} right)

 

=3×2sin⁡(3π2)sin⁡(π4)= 3 times 2sin left( dfrac{3pi}{2}right) sin left( dfrac{pi}{4} right)

 

=3(cos⁡5π4−cos⁡7π4)= 3left( cos dfrac{5pi}{4} – cos dfrac{7pi}{4}right)

 

=3(−12−12)=−32= 3left( dfrac{-1}{sqrt{2}} – dfrac{1}{sqrt{2}}right) = -3sqrt{2}

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Question 3: What are the minimum and maximum values of 3cos⁡x+4sin⁡x3cos x + 4sin x?

[2 marks]

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We have a=3a = 3 and b=4b = 4.

Then, we have R2=a2+b2R^2 = a^2 + b^2, or, R=a2+b2=25=5R = sqrt{a^2 + b^2} = sqrt{25} = 5.

Therefore, the minimum and maximum values of 3cos⁡x+4sin⁡x3cos x + 4sin x must be −5-5 and 55.

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Additional Resources

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Specification Points Covered

E6 – Understand and use double angle formulae; use of formulae for sin⁡(A±B)sin{(Apm B)}, cos⁡(A±B)cos{(Apm B)} and tan⁡(A±B)tan{(Apm B)}; understand geometrical proofs of these formulae

Understand and use expressions for acos⁡θ+bsin⁡θacos{theta}+bsin{theta} in the equivalent forms of rcos⁡(θ±α)rcos{(theta pm alpha)} or rsin⁡(θ±α)rsin{(theta pm alpha)}

R Addition Formulae Worksheet and Example Questions

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Addition and Double Angle Formulae

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