For this matrice,
[[0.9cos(-30), 0.9cos(60), 2]
[0.9sin(30), 0.9sin(60), 4]
[0, 0, 1]]
What is the rotation (in degrees) and is it clockwise or counterclockwise?
I think it's 30 degrees clockwise...Is that right?
To determine the rotation angle and direction, we can use the method of trigonometry in 2D transformations.
Given the matrix:
[[0.9cos(-30), 0.9cos(60), 2]
[0.9sin(30), 0.9sin(60), 4]
[0, 0, 1]]
Let's focus on the elements in the top row:
0.9cos(-30) and 0.9cos(60)
From trigonometry, we know that cos(-θ) = cos(θ) and cos(60) = 0.5.
Therefore, 0.9cos(-30) = 0.9cos(30) and 0.9cos(60) = 0.9*0.5 = 0.45.
Now let's focus on the elements in the second row:
0.9sin(30) and 0.9sin(60)
From trigonometry, we know that sin(-θ) = -sin(θ) and sin(60) = √3/2.
Therefore, 0.9sin(30) = 0.9*sin(30) and 0.9sin(60) = 0.9*(√3/2) ≈ 0.78.
The elements in the third column [2, 4, 1] indicate the translation components. We don't need to consider them when determining the rotation.
Now, let's find the rotation angle:
Using the above calculations, we have:
cos(θ) = 0.45 and sin(θ) = 0.78.
To find the rotation angle (θ), we can use the inverse trigonometric functions:
θ = arctan(sin(θ) / cos(θ)).
θ = arctan(0.78 / 0.45) ≈ 59.04 degrees.
Therefore, the rotation angle is approximately 59.04 degrees.
To determine the direction of rotation, we need to consider the sign of the sine function.
Since the sine function is positive for angles in the first and second quadrants, the rotation is counterclockwise.
Therefore, the rotation is approximately 59.04 degrees counterclockwise.
So, your initial guess of 30 degrees clockwise is incorrect. The correct answer is approximately 59.04 degrees counterclockwise.