Rewrite 6 sin ( x ) − 5 cos ( x ) as A sin ( x + ϕ )

Question

phi should be in the interval -pi < phi < pi

Answer 1

Hi China,

An addition identity in Trigonometry that will help you is

sin(α+β) = sinα cosβ + cosα sinβ

For this case, let α = x and β = φ.

Then, we can write

sin(x+φ) = sinx cosφ + cosx sinφ

Let's now multiply through by a constant A.

Asin(x+φ) = Asinx cosφ + Acosx sinφ

For this equation to match the given, we need:

Acosφ = 6 (*)

Asinφ = -5

Divide these equations

(Asinφ)/(Acosφ) = -5/6

A cancels out and sinφ / cosφ = tanφ = -5/6

6

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y

Imagine a right triangle like the one above (not drawn to scale) in the 4th quadrant.

φ = arctan(-5/6)

Pick Eq. (*) above:

Acosφ = 6

Looking at the triangle, cosφ = 6/√(61), where the hypotenuse is √(5² + 6²) = √(25 + 36) = √(61)

So, Eq (*) becomes

A*6/√(61) = 6

Then, A = √(61)

Putting it all together, we have

√(61) sin[x + arctan(-5/6)] = 6sinx - 5cosx