# The R Formula. How to write an expression in terms of sine or cosine only.

Updated on August 31, 2011

If you have an equation written in terms of sine and cosine, then the R formula can be used to rewrite the equation in terms of sine or cosine only. The 4 forms that you are rewriting the equations in are:

asinѲ + bcosѲ = Rsin(Ѳ + α)

asinѲ - bcosѲ = Rsin(Ѳ - α)

acosѲ +bsinѲ = Rcos (Ѳ - α)

acosѲ -bsinѲ = Rcos (Ѳ + α)

where R is a positive constant and α is a value between 0⁰ and 90⁰. Let’s take a look at a couple examples of using the R formula.

Example 1

Express √3sinѲ – cosѲ in the form Rsin(Ѳ-α), R>0 and 0≤α<90⁰, and solve the equation √3sinѲ – cosѲ = 0.5 in the range 0≤Ѳ<360⁰.

First of all use the addition formula,

sin(A-B) = sinAcosB – cosAsinB,

to expand Rsin(Ѳ-α):

Rsin(Ѳ-α) = R[sinѲcosα – cosѲsinα]

= RsinѲcosα – RcosѲsinα

So we now make this equal to √3sinѲ – cosѲ:

√3sinѲ – cosѲ = RsinѲcosα – RcosѲsinα

Now comparing coefficients on both sides of the equation you can write down 2 new equations:

Rcosα = √3 (Equation A)

Rsinα = 1 (Equation B)

Now if you square both of these equations and add both equations together you can find the value of R:

R²cos²α + R²sin²α = (√3)² + 1²

R²(cos²α + sin²α) = 4

Now you can get rid of the cos²α + sin²α as this is equal to 1 (using the identity cos²A + sin²A = 1)

R² = 4

R = 2

A quick way to work out R, is to square the two numbers from equations A and B, add them together, and take the square root (√[(√3)² + 1²] = √4 = 2 (a method similar to Pythagoras)

Now sub R= 2 back into Rcosα or Rsinα to find α:

2sinα = 1 (divide both sides by 2)

sinα = 0.5 (sin inverse both sides)

α = 30⁰

So √3sinѲ – cosѲ = 2sin(Ѳ-30)

Now we have written this in the R form we can now solve the equation:

2sin(Ѳ-30) = 0.5 (divide both sides by 2)

sin(Ѳ-30) = 0.25 (sin inverse both sides)

Ѳ - 30 = 14.5⁰, 165.5⁰ (make sure you find the other solution in the required range by subtracting the 14.5⁰ from 180⁰, and then add 30⁰ to both solutions)

Ѳ = 44.5⁰ or 195.5⁰.

Lets take a look at one more example of using the R formula:

Example 2

Write 5cosѲ + 3sinѲ in the form R(cosѲ-α).

First of all use the difference formula for cosine to expand R(cosѲ-α) ,

cos(A-B) = cosAcosB + sinAsinB so

Rcos(Ѳ-α) = R[cosѲcosα + sinѲsinα]

= RcosѲcosα + RsinѲsinα

So we now make this equal to 5cosѲ + 3sinѲ:

5cosѲ + 3sinѲ = RcosѲcosα + RsinѲsinα

Now comparing coefficients on both sides of the equation you can write down 2 new equations:

Rcosα = 5 (Equation A)

Rsinα = 3 (Equation B)

Now if you square both of these equations and add both equations together you can find the value of R or you can use the quick Pythagoras method as shown above.

√(5² + 3²) = √34

Now sub R= √34 back into Rcosα = 5 or Rsinα = 3 to find α:

√34sinα = 3 (divide both sides by 3)

sinα = 3/√34 (sin inverse both sides)

α = 31.0⁰

Therefore, 5cosѲ + 3sinѲ = √34(cosѲ-31.0) which you can now go on to use to solve equations.

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