Question

a ray on the positive x-axis is rotated -450°. what is the cosine of the angle formed by this rotation? (1 point)
$\frac{sqrt{2}}{2}$
0
-1
1

Explanation:

Step1: Find coterminal angle

To find the coterminal angle of \(-450^\circ\), we add multiples of \(360^\circ\) until we get an angle between \(0^\circ\) and \(360^\circ\).
\(-450^\circ + 360^\circ=-90^\circ\)
\(-90^\circ+ 360^\circ = 270^\circ\)? Wait, no, wait: Wait, \(-450^\circ+ 2\times360^\circ=-450 + 720 = 270^\circ\)? Wait, no, let's do it correctly. The formula for coterminal angles is \(\theta + 360^\circ n\), where \(n\) is an integer. We want to find \(n\) such that \(\theta+360n\) is between \(0\) and \(360\). For \(\theta=-450\), let's solve \(-450 + 360n\in[0,360]\). Let's try \(n = 2\): \(-450+720 = 270\)? No, wait \(n = 1\): \(-450 + 360=-90\), \(n = 2\): \(-450+720 = 270\), \(n = 3\): \(-450 + 1080=630\) (too big). Wait, actually, the reference angle or coterminal angle: Wait, \(-450^\circ\) is equivalent to \(-450^\circ+ 2\times360^\circ=270^\circ\)? No, wait, \(-450^\circ\) is a clockwise rotation. Let's think in terms of standard position. A negative angle means clockwise rotation. So \(-450^\circ\) is the same as rotating clockwise \(450^\circ\), which is the same as rotating clockwise \(450 - 360=90^\circ\) more than a full circle (since \(360^\circ\) is a full circle). Wait, no: \(450^\circ=360^\circ + 90^\circ\), so clockwise \(450^\circ\) is the same as clockwise \(90^\circ\) (because \(360^\circ\) clockwise is the same as starting over). Wait, no: clockwise \(360^\circ\) brings us back to the positive x - axis. Then clockwise \(90^\circ\) more (since \(450 = 360+90\)) would bring us to the negative y - axis. Wait, \(-450^\circ\) is equal to \(-450^\circ+ 2\times360^\circ=270^\circ\)? Wait, \(270^\circ\) is a counter - clockwise rotation of \(270^\circ\), which is the same as clockwise \(90^\circ\) (because \(360 - 270=90\)). So the angle \(-450^\circ\) is coterminal with \(270^\circ\)? Wait, no, let's calculate the coterminal angle correctly. The formula for coterminal angles is \(\alpha=\theta + 360k\), \(k\in\mathbb{Z}\), such that \(0\leq\alpha<360\). For \(\theta=-450\), we solve \(-450 + 360k\geq0\). \(360k\geq450\), \(k\geq\frac{450}{360}=\frac{5}{4} = 1.25\). So the smallest integer \(k\) is \(2\). Then \(\alpha=-450+360\times2=-450 + 720 = 270^\circ\). So the angle \(-450^\circ\) is coterminal with \(270^\circ\).

Step2: Find \(\cos(270^\circ)\)

We know the unit - circle definition of cosine: for an angle \(\theta\) in standard position, \(\cos\theta=x\), where \((x,y)\) is the point on the unit circle corresponding to the angle \(\theta\).
For \(\theta = 270^\circ\), the point on the unit circle is \((0,-1)\). So \(\cos(270^\circ)=0\).

Answer:

\(0\) (corresponding to the option "0")