Please help explain how to graph Cotangent and Sin x + cos x
For cotangent use the values for $\displaystyle \displaystyle\frac{\pi}{6}, \ \frac{\pi}{4}, \ \frac{\pi}{3}, \ \frac{\pi}{2}, \ \frac{2\pi}{3}, \ \frac{3\pi}{4}, \ \frac{5\pi}{6}$ and the vertical asympthotes.
For the second one we have
$\displaystyle \sin x+\cos x=\sin x+\sin\left(\frac{\pi}{2}-x\right)=2\sin\frac{\pi}{4}\cos\left(\frac{\pi}{4}-x\right)=\sqrt{2}\cos\left(\frac{\pi}{4}-x\right)$
1) to graph cot(x).
You should know how to graph tan(x).
cot(x) = tan(90 -x) ----in degrees.
cot(x)
= tan(90 -x)
= tan[-(x -90)]
= -tan(x -90) ------the tangent of a negative angle is the negative value of the tangent of that angle. View the angle to be in the 4th quadrant, where tangent is negative.
So, you graph -tan(x -90) instead of cot(x).
That is a basic tangent curve that is shifted 90 degrees to the right, and it is "inverted" because of the negative sign.
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2) to graph sin(x) +cos(x)
cos(x) = sin(90 -x) ----in degrees.
cos(x)
= sin(90 -x)
= sin[-(x -90)]
= -sin(x -90) -----same reasoning as in tan[-(x -90)].
So,
sin(x) +cos(x)
= sin(x) -sin(x -90)
Using the trig identity
sinA -sinB = 2cos[(A+B)/2]sin[(A-B)/2],
= 2cos[(x +x -90)/2]sin[(x -x +90)/2]
= 2cos(x -45)sin(45)
= 2cos(x -45)*[1 / sqrt(2)]
= sqrt(2)cos(x -45) --------------***
So graph sqrt(2)cos(x -45) instead of sin(x) +cos(x).
That is a cosine curve that has an amplitude of sqrt(2), and a horizontal shift of 45 degrees to the right.