Question

Как понять когда надо вычислять синус угла между векторами,а когда косинус?

Answer 1

Синус между векторами a и b  определяется при векторном произведении: его модуль численно равен площади параллелограмма, построенного на векторах  a и b. Надо еще помнить, что sinα   =  cos(90-α)

Answer 2

Understanding When to Calculate the Sine and Cosine of the Angle Between Vectors

When determining whether to calculate the sine or cosine of the angle between vectors, it's important to consider the specific context and the mathematical relationships involved.

Sine of the Angle Between Vectors: - The sine of the angle between two vectors is often used when the focus is on the perpendicular component of one vector with respect to the other. - In geometric terms, the sine of the angle between vectors can be used to find the length of the projection of one vector onto the other.

Cosine of the Angle Between Vectors: - The cosine of the angle between vectors is commonly employed when the emphasis is on the parallel component of one vector with respect to the other. - Geometrically, the cosine of the angle between vectors can be used to find the length of the adjacent side of the triangle formed by the two vectors.

In summary, the choice between calculating the sine or cosine of the angle between vectors depends on whether the focus is on the perpendicular or parallel components of the vectors in relation to each other.

Answer 3

Understanding When to Calculate the Sine and Cosine of the Angle Between Vectors

When determining whether to calculate the sine or cosine of the angle between vectors, it's important to consider the specific context and the mathematical operations involved. Here's a detailed explanation based on the provided search results:

1. Magnitude and Direction: - When dealing with vectors, the sine and cosine functions are often used to calculate the angle between vectors based on their magnitudes and directions.

2. Sine Function: - The sine function is commonly used when the angle between vectors needs to be determined based on their magnitudes and the perpendicular distance between them. - It's particularly useful when the focus is on the perpendicular components of the vectors and their relative orientation in space.

3. Cosine Function: - On the other hand, the cosine function is employed when the angle between vectors is to be calculated based on their magnitudes and the parallel distance between them. - It's often used to determine the angle between vectors when the emphasis is on their parallel components and their alignment in space.

4. Practical Application: - In practical terms, the decision to use the sine or cosine function for calculating the angle between vectors depends on the specific problem at hand, such as the nature of the vectors and the geometric relationships involved.

5. Mathematical Representation: - The mathematical representation of the angle between vectors using the sine and cosine functions is based on the fundamental principles of trigonometry and vector operations.

In summary, the choice between calculating the sine or cosine of the angle between vectors depends on the specific geometric and mathematical considerations involved in the problem, such as the perpendicular and parallel components of the vectors and their relative orientations in space.