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The auxiliary-angle method

Properties1
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The second method for linear equations sums asin⁡x+bcos⁡xa\sin x + b\cos x into a single sine function, with suitable amplitude and phase, reducing the equation to an elementary one. It is often preferable to the parametric method because it does not require checking the case x=πx=\pi separately.

In summary — Auxiliary-angle procedure

  1. Write asin⁡x+bcos⁡x=−ca\sin x + b\cos x = -c.
  2. Divide both sides by a2+b2\sqrt{a^2+b^2}: aa2+b2sin⁡x+ba2+b2cos⁡x=−ca2+b2.\frac{a}{\sqrt{a^2+b^2}}\sin x + \frac{b}{\sqrt{a^2+b^2}}\cos x = -\frac{c}{\sqrt{a^2+b^2}}.
  3. Find an angle φ\varphi such that cos⁡φ=aa2+b2\cos\varphi = \dfrac{a}{\sqrt{a^2+b^2}} and sin⁡φ=ba2+b2\sin\varphi = \dfrac{b}{\sqrt{a^2+b^2}}. Such a φ\varphi always exists, because the two values are the cosine and sine of one and the same angle (the sum of their squares equals 11).
  4. The equation becomes cos⁡φsin⁡x+sin⁡φcos⁡x=−ca2+b2\cos\varphi\sin x + \sin\varphi\cos x = -\dfrac{c}{\sqrt{a^2+b^2}}, that is sin⁡(x+φ)=−ca2+b2\sin(x+\varphi) = -\dfrac{c}{\sqrt{a^2+b^2}}: an elementary equation.

Example — The same example as before, with the auxiliary angle

3sin⁡x+cos⁡x+1=0\sqrt{3}\sin x + \cos x + 1 = 0.

Here a2+b2=3+1=2\sqrt{a^2+b^2} = \sqrt{3+1} = 2. Dividing: 32sin⁡x+12cos⁡x=−12.\frac{\sqrt{3}}{2}\sin x + \frac{1}{2}\cos x = -\frac{1}{2}. We look for φ\varphi with cos⁡φ=3/2\cos\varphi = \sqrt{3}/2 and sin⁡φ=1/2\sin\varphi = 1/2: it is φ=π/6\varphi = \pi/6 (first quadrant, cosine and sine positive). Hence: sin⁡(x+π6)=−12.\sin\left(x+\frac{\pi}{6}\right) = -\frac{1}{2}. The solutions: x+π6=−π6+2kπx+\dfrac{\pi}{6} = -\dfrac{\pi}{6} + 2k\pi or x+π6=π+π6+2kπx+\dfrac{\pi}{6} = \pi + \dfrac{\pi}{6} + 2k\pi. That is: x=−π3+2kπ ∨ x=π+2kπ.x = -\frac{\pi}{3} + 2k\pi \ \vee\ x = \pi + 2k\pi. The same result as the parametric method, but without having to check separately the “lost” case x=πx=\pi, which here appears naturally. Precisely for this reason the auxiliary-angle method is often preferable.

Vector interpretation: dividing by a2+b2\sqrt{a^2+b^2} brings the vector (a,b)(a,b) onto the unit circle, giving the cosine and sine of the auxiliary angle φ\varphi.

Topics: Trigonometric equations
Concepts: Auxiliary angle · Elementary equation · Linear trigonometric equation
Functions: Cosine · Sine
Methods: Trigonometric equations
Skills: Solving equations · Using formulae

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