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vec(u)=(5,8) Find the direction angle of vec(u). Enter your answer as an angle in degrees between 0^(@) and 360^(@) rounded to the nearest hundredth. theta=◻" 。 "

u⃗=(5,8) \vec{u}=(5,8) \newlineFind the direction angle of u⃗ \vec{u} . Enter your answer as an angle in degrees between 0∘ 0^{\circ} and 360∘ 360^{\circ} rounded to the nearest hundredth.\newlineθ=□∘ \theta=\square^{\circ}

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Inverses of sin, cos, and tan: degreesright chevron icon

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Q. u⃗=(5,8) \vec{u}=(5,8) \newlineFind the direction angle of u⃗ \vec{u} . Enter your answer as an angle in degrees between 0∘ 0^{\circ} and 360∘ 360^{\circ} rounded to the nearest hundredth.\newlineθ=□∘ \theta=\square^{\circ}
  1. Use Arctangent Function: To find the direction angle of the vector u⃗=(5,8)\vec{u} = (5,8), we need to use the arctangent function, which gives us the angle whose tangent is the ratio of the yy-coordinate to the xx-coordinate of the vector. The formula for the direction angle θ\theta is θ=arctan⁡(yx)\theta = \arctan(\frac{y}{x}), where yy is the yy-coordinate and xx is the xx-coordinate of the vector.
  2. Calculate Ratio: First, we calculate the ratio of the y-coordinate to the x-coordinate of the vector u⃗=(5,8)\vec{u} = (5,8). This ratio is 85\frac{8}{5}.
  3. Find Angle: Next, we use the arctangent function to find the angle. θ=arctan⁡(85)\theta = \arctan(\frac{8}{5}). We need to ensure that our calculator is set to degree mode since we want the answer in degrees.
  4. Calculate Arctangent: After calculating the arctangent of 85\frac{8}{5}, we get θ≈arctan⁡(1.6)\theta \approx \arctan(1.6). Using a calculator, we find that θ≈58.00\theta \approx 58.00 degrees. This is the direction angle of the vector in the first quadrant, which is between 0∘0^\circ and 90∘90^\circ.
  5. Check Quadrant: Since the vector u⃗=(5,8)\vec{u} = (5,8) is in the first quadrant and we are looking for an angle between 0∘0^\circ and 360∘360^\circ, the direction angle we found is already in the correct range. Therefore, we do not need to adjust the angle.

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