Usually I am able to find the equation of the tangent when given at least the x point... But in case, I just got the domain.
This is my problem of finding the equation of a tangent line of the graph:
$ {-\sin}^2x + {\frac{1}{2}}, x \in [0, {\frac{\pi}{2}}] $ which makes an angle of 135◦ with the x−axis. Assume that the scales along the x− axis and y− axis are the same. Angles are measured anti-clockwise from the positive x−axis.
Which point should I take as the x coordinate... $0$ or $\frac{\pi}{2}$ ?
I would really appreatiate your help, and some explanation, please.
Help has been given in the comments.
$$ y= \frac12 -\sin^2 x $$
$$ \frac{dy}{dx}= - \sin 2x = \tan 135^{\circ}= -1 $$
$$ 2x= \frac{\pi}{2},\frac{5\pi}{2} ;\,$$
$$ x= \frac{\pi}{4}, \,y=0, $$
$$ x= \frac{5\pi}{4}, \,y=0. $$
among other c0-terminal angles.
The first tangent has equation (slope at given point):
$$ \frac{y}{x-\pi/4} =-1,$$
The second tangent has equation:
$$ \frac{y}{x-5\pi/4} =-1,$$
&c.