Final answer:
The vector w = -45i + 28j in trigonometric form, rounded to the nearest degree, is w = 53cos(148°)i + 53sin(148°)j, representing the vector's magnitude and direction.
Step-by-step explanation:
The student's question revolves around converting a vector given in Cartesian form into its trigonometric form. In order to find the trigonometric form of the vector w = -45i + 28j, we need to calculate both the magnitude of the vector and the angle it makes with the positive x-axis. The magnitude can be found using the Pythagorean theorem:
√((-45)^2 + (28)^2) = √(2025 + 784) = √(2809) = 53.
Next, we calculate the angle θ using the arc tangent function based on the components of the vector:
θ = atan2(28, -45) ≈ atan2(j, i) = atan2(opposite, adjacent).
Since the vector is in the second quadrant due to the negative i (x) component and positive j (y) component, we calculate the angle with respect to the positive x-axis:
θ = 180° + tan¹(28 / -45) ≈ 180° - tan¹(28 / 45) ≈ 148° (rounded to the nearest degree).
Now we can write vector w in trigonometric form:
w = 53cos(148°)i + 53sin(148°)j.
Therefore, the trigonometric form that represents vector w is w = 53cos(148°)i + 53sin(148°)j.