Rotated Image
Description
Typically, canvas appears static, thus we must resize images to fit it. However, this one is different. The canvas must be reshaped (with a set ratio) to fit the static-sized image.
You have a rectangular image with size $a \times b$ and a canvas of height and width ratio $m:n$. What would be the smallest canvas size that would fit the image?
Isn't this too simple?
Play a bit more. The image is rotated by $\theta$ degrees clockwise. Now, determine the smallest canvas size needed to fit the image using that ratio.
A rotated image inside a canvas. Note: $m, n$ may NOT be co-prime.
Each test contains multiple test cases. The first line contains the number of test cases T ($1\leq T \leq 10^5$). The description of the test cases follows.
Each test case consists of a single line containing five integers $a, b, m, n$, and $\theta$ ($1 \leq a,b,m,n \leq 10^9, 0 \leq \theta \leq 90$) — image size $a\times b$, canvas ratio $m:n$ and rotation of the image with the horizontal line.
For each test case, output two integers — the minimum size of the canvas X, Y — height and width. Answer must be integer value.
Input
Each test contains multiple test cases. The first line contains the number of test cases T ($1\leq T \leq 10^5$). The description of the test cases follows.
Each test case consists of a single line containing five integers $a, b, m, n$, and $\theta$ ($1 \leq a,b,m,n \leq 10^9, 0 \leq \theta \leq 90$) — image size $a\times b$, canvas ratio $m:n$ and rotation of the image with the horizontal line.
Output
For each test case, output two integers — the minimum size of the canvas X, Y — height and width. Answer must be integer value.
1
6 18 2 3 30
16 24
Note
Dataset is huge, use faster I/O methods.