Give one value of global angle θ of a vector, for which the x and y-components of the vector will have the same magnitude.
θ = 45 degrees
tan θ = y/x = 1
So, where is that true?
To find a value of the global angle θ for which the x and y-components of a vector have the same magnitude, we can use the trigonometric properties of right triangles. Let's assume the magnitude of both the x and y-components is equal to a value, say "d."
In a right triangle, the x-component is the adjacent side and the y-component is the opposite side. The tangent (tan) function relates the opposite and adjacent sides of a right triangle.
The tangent of an angle θ is defined as the ratio of the length of the opposite side to the length of the adjacent side, which can be represented as:
tan(θ) = opposite/adjacent= y-component/x-component
Since we want the magnitude of the x and y-components to be the same, we set the y-component equal to the x-component:
d/d = 1
Thus, tan(θ) = 1.
To find a value of θ for which the tangent is 1, we can use trigonometric identities. The tangent function is positive in both the first and third quadrants, and in the first and third quadrants, the tangent takes on the value of 1 when θ equals 45 degrees or π/4 radians.
So, one value of the global angle θ for which the x and y-components of the vector will have the same magnitude is 45 degrees or π/4 radians.
To find the value of the global angle θ for which the x and y-components of a vector have the same magnitude, we can start by considering the components of the vector.
Let's assume the magnitude of the x-component is given by Ax and the magnitude of the y-component is given by Ay. Since we want these magnitudes to be equal, we can write:
Ax = Ay
Now, we can use the trigonometric definitions of the x and y-components to relate them to the magnitude of the vector and the angle θ. The x-component can be written as:
Ax = A * cos(θ)
Similarly, the y-component can be written as:
Ay = A * sin(θ)
Substituting these expressions into the equation Ax = Ay, we get:
A * cos(θ) = A * sin(θ)
Dividing both sides of the equation by A, we have:
cos(θ) = sin(θ)
Now, we need to find the values of θ for which the cosine and sine functions are equal.
One such value is θ = 45° (or π/4 radians). At this angle, the cosine and sine functions are both equal to √2/2, making the x and y-components of the vector have the same magnitude.
So, one value of θ for which the x and y-components of the vector have the same magnitude is θ = 45° (or π/4 radians).