Auxiliary Angle/Examples

Examples of Auxiliary Angles

$3 \cos x$ minus $2 \sin x$

$3 \cos x - 2 \sin x = \sqrt {13} \map \cos {x + \arctan \dfrac 2 3}$

Hence the greatest value of $3 \cos x - 2 \sin x$ is $\sqrt {13}$ which happens when $x = -\arctan \dfrac 2 3$.

Solutions to $3 \cos x - 2 \sin x = 1$

Consider the equation:

$(1): \quad 3 \cos x - 2 \sin x = 1$

The solutions to $(1)$ between $0 \degrees$ and $360 \degrees$ are:

\(\ds x\)

\(=\)

\(\ds 40 \degrees \, 20'\)

\(\ds x\)

\(=\)

\(\ds 252 \degrees \, 24'\)