Given that θ is an angle in standard position whose terminal side contains
the given point, provide the exact value of the indicated function.
(-2, -3) Find sec θ.
construct your triangle in the third quadrant,
using Pythagoras, you know
r^2 = (-2)^2 + (-3)^2 = 13
r = √13
then cosØ = x/r = -2/√13 , then
sec Ø = -√13/2
To find the exact value of sec θ, we need to determine the values of the adjacent side and the hypotenuse in the right triangle formed by the point (-2, -3).
First, let's find the distance from the origin (0,0) to the given point (-2, -3) using the Pythagorean theorem:
d = √((-2)^2 + (-3)^2)
d = √(4 + 9)
d = √13
Next, let's determine the signs of the coordinates (-2, -3). Since -2 is negative and -3 is negative, we are in the third quadrant.
In the third quadrant, the adjacent side is represented by the x-coordinate (-2), and the hypotenuse is represented by the distance from the origin (√13).
The secant function is defined as the reciprocal of the cosine function: sec θ = 1/cos θ.
To find the cosine of θ, we divide the adjacent side (which is -2) by the hypotenuse (√13):
cos θ = -2 / √13
Now, we can find the secant of θ by taking the reciprocal of the cosine of θ:
sec θ = 1 / cos θ = 1 / (-2 / √13)
sec θ = √13 / -2
Therefore, the exact value of sec θ for the given point (-2, -3) is -√13 / 2.