Question

The​ equation, with a restriction on​ x, is the terminal side of an angle theta in standard position. y= -2x Quadrant IV.

Answer 1

In Q4, the terminal ray of our reference angle is the hypotenuse of the right triangle created by connecting the end of the terminal ray to the positive x-axis.  Theta is the angle between these 2.  In the equation, in the simplest form, if we sub in a 1 for x, then y = -2.  That means the distance along the x axis (the base of the right triangle) is 1, and the height of the triangle is -2.  We need now to find the measure of the hypotenuse using Pythagorean's Theorem.  $c^2=1^2+(-2)^2$  and  $c^2=5$  so  $c= \sqrt{5}$.  The sin of theta is the ratio of the side opposite over the hypotenuse, so $sin \theta =- \frac{2}{ \sqrt{5} }$,  and $csc \theta =- \frac{ \sqrt{5} }{2}$.  $cos \theta= \frac{1}{ \sqrt{5} }$  and $sec \theta = \frac{ \sqrt{5} }{1}$.  $tan \theta = -\frac{2}{1}$  so  $cot \theta =- \frac{1}{2}$.  There you go!