Question

Express each of the following in the form r sin(x+a), where r>0 and 0<a<2pi.
(a )5cosx+12sinx
(b) 12cosx+5sinx

Answer 1

Express each of the following in the form $r \sin(x+a)$, where $r > 0$ and $0 < a < 2pi$.

(a) $5\cos x + 12\sin x$

(b) $12\cos x + 5\sin x$

Solution:

$$5^2 + 12^2 = 13^2$$ $$\left(\frac{5}{13}\right)^2 + \left(\frac{12}{13}\right)^2 = 1$$

(a) $\sin a = \frac{5}{13}$, $\cos a = \frac{12}{13}$

$$\sin^2 a + \cos^2 a = 1$$ $$5\cos x + 12\sin x = 13\left(\frac{5}{13}\cos x + \frac{12}{13}\sin x\right) = 13\left(\sin a\cos x + \cos a\sin x\right) = 13\sin(x + a)$$ $$(b)\sin a = \frac{12}{13}, \cos a = \frac{5}{13}$$ $$12\cos x + 5\sin x = 13\left(\frac{12}{13}\cos x + \frac{5}{13}\sin x\right) = 13\left(\sin a\cos x + \cos a\sin x\right) = 13\sin(x + a)$$

Answer: (a) $13\sin(x + a)$, (b) $13\sin(x + a)$.