Final answer:
To find the acute angle of intersection between two curves at the origin, calculate the gradients at the intersection point, use the tangent formula involving the gradients, take the inverse tangent, and ensure the angle is in radians.
Step-by-step explanation:
To find the acute angle of intersection between two curves in radians, we consider the gradients (slopes) of these curves at their point of intersection. If m1 and m2 are the gradients of the two curves at the intersection point, then the tangent of the angle θ between them is given by the formula:
tan(θ) = |(m2 - m1) / (1 + m1*m2)|
Once tan(θ) is known, we can find θ by taking the inverse tangent (arctan or tan-1). In radians, there are 2π in a 360° rotation, so we use this relationship to convert from degrees to radians if necessary.
As an example, if one curve has a gradient of 2 and another has a gradient of -1/2 at their intersection (and assuming they intersect at the origin), the angle of intersection in radians can be found as follows:
- Calculate the tangent of the angle using the gradients: tan(θ) = |(-1/2 - 2) / (1 + 2*-1/2)|.
- Simplify the fraction to find tan(θ).
- Find θ by taking tan-1(θ), ensuring the angle is acute.
- Convert θ to radians if it's not already in that unit.
Assuming the calculations are carried out correctly, this process will yield the acute angle of intersection in radians.