If θ is an angle in standard position and its terminal side passes through the point (4,9), find the exact value of secθ in simplest radical form.
To find the exact value of secθ, we need to determine the value of the cosine of θ since secθ is the reciprocal of cosine. We can find the cosine of θ by using the coordinates of the point (4, 9) on the terminal side of the angle in standard position.
The given point (4, 9) represents the coordinates (x, y) on the Cartesian plane. To find the distance r between the point and the origin (0, 0), we can use the Pythagorean theorem, which states that for any right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the distance r is the hypotenuse, and the other two sides are x and y. We can calculate r using the Pythagorean theorem as follows:
r^2 = x^2 + y^2
r^2 = 4^2 + 9^2
r^2 = 16 + 81
r^2 = 97
Taking the square root of both sides, we find:
r = √(97)
Now that we have the value of r, we can find the cosine of θ. Since cosine is the ratio of the adjacent side (x) to the hypotenuse (r), we can calculate it as:
cosθ = x / r
cosθ = 4 / √(97)
Finally, we can find the exact value of secθ by taking the reciprocal of cosine:
secθ = 1 / cosθ
secθ = 1 / (4 / √(97))
secθ = √(97) / 4
So, the exact value of secθ in simplest radical form is √(97) / 4.
you have
x = 4
y = 9
r = √97
secθ = r/x