Question

Find the value of the cosecant of an angle given that the point on the terminal side of an angle is (3,4)

Answer 1

Solution:

Using the point (3,4), we can see that this forms a right triangle that has a base that is 3 units in length and an adjoining leg that is 4 units high. We are able to find the hypotenuse of this triangle using the Pythagorean Theorem. This will give us the distance of the point (3,4) to the origin.

So, $a=3,b=4$

Using,

$a^2+b^2=c^2 \\ \Rightarrow 3^2+4^2=c^2 \\\Rightarrow c^2=9+16=25 \\\Rightarrow c=5$

And so the hypotenuse of this triangle (the distance from our point we are working with to the origin), is 5 units long. Recall that when using cosecant for right triangles, cosecant represents the following

$\cosec=\dfrac{Hypotenuse}{Perpendicular}=\dfrac cb \\\Rightarrow \cosec=\dfrac 54$