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My problem says to find the measure of each acute angle $\theta$ to the nearest degree.

$$\large\cos\theta = 0.2249$$

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Basically you have to apply the inverse function of $\cos$ on both sides ( ) the problem is that $\cos$ is not one-to-one therefore you have to specify the interval of theta you are looking for. – Listing Feb 18 '12 at 17:12… – Gigili Feb 18 '12 at 17:14
up vote 4 down vote accepted

There are two places on the unit circle corresponding to values of $\theta$ satisfy your equation:

Mathematica graphics

The inverse cosine (or arccosine) function will give us the one that's in the range $[0,\pi]$: $$\arccos(0.2249)\approx1.34396.$$ The other location can be described by the opposite of that value. We can also add any integer multiple of $2\pi$ to either of these values and still be at the same place on the unit circle, with the same value for cosine, so: $$\theta=\pm\arccos(0.2249)+2\pi k,\quad k\in\mathbb{Z}.$$

edit Oops, you only asked for the acute one, and in degrees. The arccosine function gives the unique result in $[0^\circ,180^\circ]$, and acute angles are in $(0^\circ,90^\circ)$, so if there is an acute angle, it will be $\theta=\arccos(0.2249)\approx77^\circ$.

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