The Correct Answer and Explanation is:
The equation is verified by carrying out the matrix multiplication and using trigonometric sum identities. The result is expected because the multiplication of 2D rotation matrices is equivalent to composing the rotations, which means their angles are added.
Verification
To verify the equation, we perform the matrix multiplication on the left-hand side (LHS):
LHS = [cos 32 -sin 32] [cos 40 -sin 40]
[sin 32 cos 32] [sin 40 cos 40]
The resulting matrix is found by calculating each element:
- Top-left element (Row 1, Column 1):
(cos 32)(cos 40) + (-sin 32)(sin 40) = cos 32 cos 40 – sin 32 sin 40
Using the cosine sum identity, cos(A + B) = cos A cos B – sin A sin B, this simplifies to cos(32 + 40) = cos 72. - Top-right element (Row 1, Column 2):
(cos 32)(-sin 40) + (-sin 32)(cos 40) = -cos 32 sin 40 – sin 32 cos 40 = -(sin 32 cos 40 + cos 32 sin 40)
Using the sine sum identity, sin(A + B) = sin A cos B + cos A sin B, this simplifies to -sin(32 + 40) = -sin 72. - Bottom-left element (Row 2, Column 1):
(sin 32)(cos 40) + (cos 32)(sin 40) = sin 32 cos 40 + cos 32 sin 40
Using the sine sum identity, this simplifies to sin(32 + 40) = sin 72. - Bottom-right element (Row 2, Column 2):
(sin 32)(-sin 40) + (cos 32)(cos 40) = cos 32 cos 40 – sin 32 sin 40
Using the cosine sum identity, this simplifies to cos(32 + 40) = cos 72.
Combining these elements gives the resulting matrix:
[cos 72 -sin 72]
[sin 72 cos 72]
This matches the right-hand side of the equation, so the identity is verified.
Explanation
This result could have been expected because the matrices involved are standard 2D rotation matrices. A matrix of the form:
R(θ) = [cos θ -sin θ]
[sin θ cos θ]
represents a linear transformation that rotates points in the Cartesian plane counterclockwise by an angle θ around the origin.
In the given problem:
- The first matrix is R(32), representing a rotation of 32 degrees.
- The second matrix is R(40), representing a rotation of 40 degrees.
In linear algebra, multiplying transformation matrices corresponds to performing the transformations in sequence (from right to left). Therefore, multiplying R(32) by R(40) represents the composite transformation of first rotating by 40 degrees, and then rotating by an additional 32 degrees.
Geometrically, a rotation of 40 degrees followed by a rotation of 32 degrees is equivalent to a single rotation by the sum of the angles, 40 + 32 = 72 degrees. The matrix for a 72-degree rotation is R(72). Thus, we can expect that R(32) * R(40) = R(32 + 40) = R(72), which is exactly what the equation states. This property makes matrix multiplication a powerful tool for describing composite geometric transformations
