Normally you use sine theta with the cross product and cos theta with the vector product, so that the cross product of parallel vectors is zero while the dot product of vectors at right angles is zero.
In vector mathematics, the cross product involves the sine of the angle because it measures the area of the parallelogram formed by the two vectors, which is maximized when the vectors are perpendicular (90 degrees) and zero when they are parallel (0 degrees). On the other hand, the dot product uses the cosine of the angle because it quantifies the extent to which one vector extends in the direction of another, achieving its maximum when the vectors are aligned (0 degrees) and zero when they are perpendicular (90 degrees). This geometric interpretation aligns with the respective relationships of sine and cosine to angles in right triangles.
The dot product measures the extent to which two vectors align in the same direction, which is directly related to the cosine of the angle between them; thus, ( \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos(\theta) ). In contrast, the cross product gives a vector that is perpendicular to the plane formed by the two vectors, and its magnitude is proportional to the sine of the angle between them; hence, ( |\mathbf{A} \times \mathbf{B}| = |\mathbf{A}| |\mathbf{B}| \sin(\theta) ). This distinction arises from the geometric interpretations of these operations in relation to the angle between the vectors.
The sine function is used in the cross product because the magnitude of the cross product of two vectors is determined by the area of the parallelogram formed by those vectors. This area is calculated as the product of the magnitudes of the vectors and the sine of the angle between them. Specifically, the formula for the cross product (\mathbf{A} \times \mathbf{B}) includes (|\mathbf{A}||\mathbf{B}|\sin(\theta)), where (\theta) is the angle between the vectors, capturing the component of one vector that is perpendicular to the other. Thus, the sine function accounts for the directional aspect of the vectors in determining the resultant vector's magnitude and orientation.
That is how the scalar product is defined. Also, the projection of one vector onto another at an angle to it is directly proportional to the cosine of that angle.
To find the angle between two vectors, you can use the dot product formula: ( \cos(\theta) = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|} ), where ( \theta ) is the angle between the vectors, ( \mathbf{A} ) and ( \mathbf{B} ) are the vectors, and ( |\mathbf{A}| ) and ( |\mathbf{B}| ) are their magnitudes. First, calculate the dot product of the two vectors, then divide by the product of their magnitudes. Finally, take the inverse cosine (arccos) of the result to find the angle in radians or degrees.
You can use your trigonometric functions (sine, cosine, and tangent).
The dot product measures the extent to which two vectors align in the same direction, which is directly related to the cosine of the angle between them; thus, ( \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos(\theta) ). In contrast, the cross product gives a vector that is perpendicular to the plane formed by the two vectors, and its magnitude is proportional to the sine of the angle between them; hence, ( |\mathbf{A} \times \mathbf{B}| = |\mathbf{A}| |\mathbf{B}| \sin(\theta) ). This distinction arises from the geometric interpretations of these operations in relation to the angle between the vectors.
The inverse (negatives) of sine, cosine, and tangent are used to calculate the angle theta (or whatever you choose to name it). Initially it is taught that opposite over hypotenuse is equal to the sine of theta sin(theta) = opposite/hypotenuse So it can be said that theta = sin-1 (opp/hyp) This works the same way with cosine and tangent In short the inverse is simply what you use when you move the sin, cos, or tan to the other side of the equation generally to find the angle
The sine function is used in the cross product because the magnitude of the cross product of two vectors is determined by the area of the parallelogram formed by those vectors. This area is calculated as the product of the magnitudes of the vectors and the sine of the angle between them. Specifically, the formula for the cross product (\mathbf{A} \times \mathbf{B}) includes (|\mathbf{A}||\mathbf{B}|\sin(\theta)), where (\theta) is the angle between the vectors, capturing the component of one vector that is perpendicular to the other. Thus, the sine function accounts for the directional aspect of the vectors in determining the resultant vector's magnitude and orientation.
That is how the scalar product is defined. Also, the projection of one vector onto another at an angle to it is directly proportional to the cosine of that angle.
The cosine function is used to determine the x component of the vector. The sine function is used to determine the y component. Consider a vector drawn on an x-y plane with its initial point at (0,0). If L is the magnitude of the vector and theta is the angle from the positive x axis to the vector, then the x component of the vector is L * cos(theta) and the y component is L * sin(theta).
To find the angle between two vectors, you can use the dot product formula: ( \cos(\theta) = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|} ), where ( \theta ) is the angle between the vectors, ( \mathbf{A} ) and ( \mathbf{B} ) are the vectors, and ( |\mathbf{A}| ) and ( |\mathbf{B}| ) are their magnitudes. First, calculate the dot product of the two vectors, then divide by the product of their magnitudes. Finally, take the inverse cosine (arccos) of the result to find the angle in radians or degrees.
pen0r
A reciprocal trigonometric function is the ratio of the reciprocal of a trigonometric function to either the sine, cosine, or tangent function. The reciprocal of the sine function is the cosecant function, the reciprocal of the cosine function is the secant function, and the reciprocal of the tangent function is the cotangent function. These functions are useful in solving trigonometric equations and graphing trigonometric functions.
You can use the Pythagorean identity to solve this:(sin theta) squared + (cos theta) squared = 1.
cosine = adjacent/hypotenuse. It can be used as other trig functions can.
Because in dot product we take projection fashion and that is why we used cos and similar in cross product we used sin