# Thread: Solving a trig equation

1. ## Solving a trig equation

Find, to the nearest degree, all values of theta in the interval 0° ≤ theta < 360°
that satisfy the equation 3 cos 2theta + sin theta – 1 = 0.

2. At:

Wolfram|Alpha

enter the command:

plot 3 cos(2x) + sin(2x) - 1 for x between 0 and 360 degrees

it looks like the equation has four zeros in your interval. Use your graphing calculator to approximate them.

Good luck!

Find, to the nearest degree, all values of $\displaystyle \theta$ in the interval $\displaystyle [0^o,\:360^o)$

that satisfy the equation: .$\displaystyle 3\cos2\theta + \sin\theta – 1 \:=\: 0$
Use the identity: .$\displaystyle \cos2\theta \:=\:1-2\sin^2\!\theta$

The equation becomes: .$\displaystyle 3(1-2\sin^2\!\theta) + \sin\theta - 1 \:=\:0 \quad\Rightarrow\quad 6\sin^2\!\theta - \sin\theta - 2 \:=\:0$

Factor: .$\displaystyle (2\sin\theta + 1)(3\sin\theta - 2) \:=\:0$

And we have two equations to solve:

. . $\displaystyle 2\sin\theta + 1 \:=\:0 \quad\Rightarrow\quad \sin\theta\:=\:\text{-}\tfrac{1}{2} \quad\Rightarrow\quad \theta \:=\:210^o,\:330^o$

. . $\displaystyle 3\sin\theta -2 \:=\:0 \quad\Rightarrow\quad \sin\theta \:=\:\tfrac{2}{3} \quad\Rightarrow\quad \theta \:\approx\:42^o,\:138^o$