If point (-5,4) is a point on the terminal side of [tex] \theta [/tex], then angle [tex] \theta [/tex] is obtuse (point (-5,4) is placed in the second quadrant). Consider right triangle with legs 5 and 4 and angle [tex] \theta_1 [/tex] ([tex] \theta_1 [/tex] is in the first quadrant) adjacent to the leg with length 5. By the Pythagorean theorem [tex] \text{ hypotenuse }^2=4^2+5^2=41 [/tex] and the length of the [tex] \text{ hypotenuse }=\sqrt{41} [/tex]. Then:
1. [tex] \sin \theta_1=\dfrac{\text{ opposite leg }}{\text {hypotenuse }} =\dfrac{4}{\sqrt{41}} [/tex];
2. [tex] \sec \theta_1=\dfrac{1}{\cos \theta_1}=\dfrac{\text{ hypotenuse }}{\text{ adjacent leg }}=\dfrac{\sqrt{41}}{5} [/tex];
3. [tex] \tan \theta_1=\dfrac{\text{opposite leg}}{\text{adjacent leg}}=\dfrac{4}{5} [/tex].
In the second quadrant function [tex] \sin [/tex] has the same sign as in the first quadrant and functions [tex] \sec, \tan [/tex] have the opposite sign.
Therefore,
1. [tex] \sin \theta=\dfrac{4}{\sqrt{41}} [/tex];
2. [tex]\sec \theta =-\dfrac{\sqrt{41}}{5} [/tex];
3. [tex]\tan \theta =-\dfrac{4}{5}[/tex].
