The general formula for writing a unit vector in terms of sin and cos is u = cos(θ)i + sin(θ)j, where θ is the angle between the unit vector and the positive x-axis. This formula is derived from the Pythagorean theorem and the definition of sine and cosine as ratios of sides in a right triangle.
Yes, you can write a unit vector in terms of only sin or cos, but it may not be a unique representation. For example, if you write u = cos(θ)i, there are infinitely many unit vectors that satisfy this equation, as long as θ is a multiple of 360 degrees.
To determine the angle θ, you can use the inverse trigonometric functions. For example, if the unit vector is u = ½i + ½j, then θ = 45 degrees, since cos(45) = sin(45) = ½.
Yes, you can also use tangent and cotangent to write a unit vector. The formula is u = cos(θ)i + sin(θ)j = cot(θ)i + tan(θ)j, where θ is the angle between the unit vector and the positive x-axis.
Yes, you can also write a unit vector in terms of its components. For example, if the unit vector is u = a i + b j, then its magnitude is |u| = √(a^2 + b^2) = 1, which can be used to find the values of a and b. This method is useful when the unit vector is not given in terms of trigonometric functions.