3 Answers:
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Explanation:
Multiply the given radian angle measure by the fraction 180/pi to convert to degree form.
This gives
[tex]\left(\frac{7\pi}{4}\right)*\left(\frac{180}{\pi}\right) = 315[/tex]
The pi terms cancel.
Therefore, 7pi/4 radians = 315 degrees.
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Use the unit circle (see below) to see that 315 degrees is in quadrant 4.
The terminal point that corresponds to this angle is [tex]\left(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right)[/tex]
This leads us to
[tex]\cos(\theta) = \frac{\sqrt{2}}{2}\\\\\sin(\theta) = -\frac{\sqrt{2}}{2}\\\\[/tex]
Since any point on the unit circle is of the form [tex](x,y) = (\cos(\theta),\sin(\theta))[/tex]
The ratio of those sine and cosine values leads to the tangent value.
[tex]\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\\\\\tan(\theta) = \sin(\theta) \div \cos(\theta)\\\\\tan(\theta) = -\frac{\sqrt{2}}{2} \div \frac{\sqrt{2}}{2}\\\\\tan(\theta) = -\frac{\sqrt{2}}{2} \times \frac{2}{\sqrt{2}}\\\\\tan(\theta) = -1\\\\[/tex]
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Go back to the 315 degree angle.
This angle is between 270 and 360, so it's in Q4.
The reference angle for anything in Q4 is 360-theta
So the reference angle here is 360-theta = 360-315 = 45.
