Answer:
Value of [tex]\sec \theta = \frac{5}{4}[/tex]
Step-by-step explanation:
Given: A vertical line is dropped from the x-axis to the point (12, -9) as shown in the diagram below;
To find the value of [tex]\sec \theta[/tex].
By definition of secants;
[tex]\sec \theta = \frac{1}{\cos \theta}[/tex]
Now, first find the cosine of angle [tex]\theta[/tex]
As the point (12 , -9) lies in the IV quadrant , where [tex]\cos \theta > 0[/tex]
Consider a right angle triangle;
here, Adjacent side = 12 units and Opposite side = -9 units
Using Pythagoras theorem;
[tex](Hypotenuse side)^2= (12)^2 + (-9)^2 = 144 + 81 = 225[/tex]
or
[tex]Hypotenuse = \sqrt{225} =15 units[/tex]
Cosine ratio is defined as in a right angle triangle, the ratio of adjacent side to hypotenuse side.
[tex]\cos \theta = \frac{Adjacent side}{Hypotenuse side}[/tex]
then;
[tex]\cos \theta = \frac{12}{15} = \frac{4}{5}[/tex]
and
[tex]\sec \theta = \frac{1}{\cos \theta}= \frac{1}{\frac{4}{5}} = \frac{5}{4}[/tex]
therefore, the value of [tex]\sec \theta[/tex] is, [tex]\frac{5}{4}[/tex]
