Express $7 \sin x - 8 \cos x$ in the form of $R \sin (x - \alpha)$, where $p, R 0$ a

Question

Express $7 \sin x - 8 \cos x$ in the form of $R \sin (x - \alpha)$, where $p, R > 0$ and $0^\circ < \alpha < 90^\circ$. Find the acute angle $x$ for which $7 \sin x - 8 \cos x = 3$. Find the minimum value of $7 \sin x - 8 \cos x$ and the corresponding value of $x$, for $0^\circ < x < 360^\circ$. the maximum value of $\frac{1}{(7 \sin x - 8 \cos x)^2 + 3}$ and the corresponding value of $x$, for $0^\circ < x < 360^\circ$.

Accepted Answer

$\sqrt{113} \sin (x - 48.814^\circ)$; (i) $x = 65.2^\circ$; (ii) minimum $= -\sqrt{113}$ at $x = 318.8^\circ$; (iii) maximum $= \dfrac{1}{3}$ at $x = 48.8^\circ$ or $228.8^\circ$

Trigonometry - Trigonometric Equations

Trigonometry - Trigonometric Equations

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