The point (7/25, 24/25) defines an angle with the x-axis. To find csc (cosecant) and cot (cotangent) for this angle, you take the reciprocals of the sine and the ratio of cosine to sine, respectively, resulting in cscθ = 25/24 and cotθ = 7/24.
The coordinates (7/25, 24/25) represent a point on the unit circle that forms an angle θ with the positive x-axis. By definition, the sine of θ is the y-coordinate, and the cosine of θ is the x-coordinate. Therefore:
From there, you can find the cosecant (cscθ), which is the reciprocal of sine, and the cotangent (cotθ), which is the reciprocal of the tangent or the ratio of cosine to sine.
To find cscθ:
- cscθ = 1/sinθ
- cscθ = 1/(24/25)
- cscθ = 25/24
To find cotθ:
- cotθ = cosθ/sinθ
- cotθ = (7/25)/(24/25)
- cotθ = 7/24
Thus, cscθ = 25/24 and cotθ = 7/24.