If θ is an angle in standard position and its terminal side passes through the point (7,-4), find the exact value of
sec
θ
secθ in simplest radical form.
r = √(7^2 + 4^2) = √65
secθ = r/x = √65/7
To find the exact value of secθ, we can use the fact that secθ is the reciprocal of cosθ.
First, let's find cosθ.
Since the terminal side of θ passes through the point (7, -4), we can create a right triangle by drawing a line segment from the origin to the point (7, -4).
The length of the horizontal leg of the triangle is 7 and the length of the vertical leg is -4.
Using the Pythagorean theorem, we can find the length of the hypotenuse:
c^2 = 7^2 + (-4)^2
c^2 = 49 + 16
c^2 = 65
c = sqrt(65)
Now, we can find cosθ:
cosθ = adjacent / hypotenuse
cosθ = 7 / sqrt(65)
Finally, to find secθ, we take the reciprocal of cosθ:
secθ = 1 / cosθ
secθ = 1 / (7 / sqrt(65))
secθ = sqrt(65) / 7
Therefore, the exact value of secθ in simplest radical form is sqrt(65) / 7.
To find the exact value of secθ, we need to first find the value of cosθ.
Given that the terminal side of θ passes through the point (7, -4), we can create a right triangle with the hypotenuse being the distance from the origin (0, 0) to the point (7, -4). Using the Pythagorean theorem, we can find the length of the hypotenuse:
h = sqrt((7)^2 + (-4)^2)
= sqrt(49 + 16)
= sqrt(65)
The value of cosθ is then given by the adjacent side divided by the hypotenuse:
cosθ = adjacent/hypotenuse
= 7/sqrt(65)
Finally, the value of secθ is the reciprocal of cosθ:
secθ = 1/cosθ
= 1 / (7/sqrt(65))
= sqrt(65)/7
Therefore, the exact value of secθ in simplest radical form is sqrt(65)/7.