If θ is an angle in standard position and its terminal side passes through the point (2,9), find the exact value of \cot\thetacotθ in simplest radical form.
cotθ = x/y = 2/9
To find the exact value of \cot\theta, we need to use the coordinates of the point (2,9) where the terminal side of the angle passes through.
Since the point (2,9) is in the first quadrant, we can use the Pythagorean theorem to find the length of the hypotenuse:
\sqrt{2^2 + 9^2} = \sqrt{4 + 81} = \sqrt{85}
We can then use the coordinates of the point to find the adjacent side:
Adjacent side = 2
Since \cot\theta = \frac{\text{Adjacent side}}{\text{Opposite side}}, and there is no given information about the opposite side, we cannot determine the exact value of \cot\theta in simplest radical form.
To find the exact value of cot θ, we need to determine the coordinates of the point where the terminal side of angle θ intersects the unit circle.
Given that the terminal side passes through the point (2, 9), we can determine the length of the hypotenuse and the adjacent side of the right triangle formed by this point.
The hypotenuse is the distance from the origin (0, 0) to the point (2, 9), which can be found using the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((2 - 0)^2 + (9 - 0)^2)
= sqrt(4 + 81)
= sqrt(85)
The adjacent side is the horizontal distance from the origin (0, 0) to the point (2, 9), which is simply 2.
Now, we can determine the value of cot θ by using the definition of cotangent:
cot θ = adjacent / opposite
In this case, the adjacent side is 2 and the opposite side is 9 (since we are dealing with a right triangle and cotangent is the ratio of the adjacent side to the opposite side).
cot θ = 2 / 9
Therefore, the exact value of cot θ is 2/9 in simplest form.