Find the exact value of csc ? for angle ? in standard position if the point (5, –12) lies on its terminal side.

Question

can you please help me with my trig question and how to get to that answer, thanks

Answer 1

The coordinate is connected to the point (0,0) and the given coordinate lies on the 4th quadrant.  The x coordinate is 5 and the y coordinate is -12.  This connectivity will form a vector.  If you draw a line from the vector's endpoint perpendicular to the x-axis, we get a right triangle.  This means that one leg of the triangle is 5, and the other leg of the triangle is 12.  We can use Pythagorean theorem to find the vector length (hypotenuse).

 

c = √(5² + 12²)

c = √(25 + 144)

c = √169

c = 13

 

Now we can find csc.

 

cscθ = 1 / sinθ

       = 1 / (y / c)

       = 1 / (-12 / 13)

       = -13 / 12

 

 

Answer 2

We know this:

 

x = 5

y = 12

 

So that means we can also know that:

 

r = √((5)² + (-12)² ) = 13

 

With r representing the distance from the given point to the origin.

 

Now that we know the origin, calculating csc(Θ) (cosecant of theta) is straightforward:

 

csc(Θ) = r / y

 

this being the ratio between the hypotenuse and the vertical side of the triangle formed between triangle formed by the line from the origin to (5, -12) and the x-axis. (note that this is the reciprocal of sine)

 

So the answer comes out as (13)/(-12) = -(13/12)