Trig question: with intervals

VkL · #1 $Old$ January 13th, 2009, 09:51 PM

Solve the equation for a in [0, 2pie]

A.) cos²(a) + 3cos(a) = -cos(a)

B.)sin(2a) = cos(a)

ThePerfectHacker · #2 $Old$ January 13th, 2009, 10:08 PM

Quote:

Originally Posted by VkL $View Post$

Solve the equation for a in [0, 2pie]

A.) cos²(a) + 3cos(a) = -cos(a)

Add $\cos a$ to both sides to get $\cos^2 a + 4\cos a = 0$ and now $\cos a ( \cos a + 4) = 0$ . Therefore, $\cos a = 0$ or $\cos a + 4 = 0$ . The second condition, $\cos a + 4 = 0 \implies \cos a = -4$ is impossible because $|\cos a| \leq 1$ . The first condition is $\cos a = 0$ . That happens when $a = \tfrac{\pi}{2},\tfrac{3\pi}{2}$

Quote:

B.)sin(2a) = cos(a)

Using the double angle identity we get $2\sin a \cos a = \cos a \implies 2\sin a \cos a - \cos a = 0$ . Now factor, $\cos a ( 2\sin a - 1) = 0$ . Thus, we get two possibilities, $\cos a = 0$ or $2\sin a - 1 = 0$ . The first condition, $\cos a = 0$ , gives $a = \tfrac{\pi}{2}, \tfrac{3\pi}{2}$ . The second condition, $2\sin a - 1 = 0$ is equivalent to saying $\sin a = \tfrac{1}{2}$ , thus, $a = \tfrac{\pi}{6}, \tfrac{5\pi}{6}$ .