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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. Calculate Tangent Ratio: To find the direction angle of the vector u⃗=(−5,−8)\vec{u} = (-5, -8), we need to calculate the angle that this vector makes with the positive x-axis. The direction angle, often denoted as θ\theta, can be found using the arctangent function (tan⁡−1\tan^{-1} or atan\text{atan}), which gives us the angle whose tangent is the ratio of the y-coordinate to the x-coordinate of the vector.
  2. Use Arctangent Function: First, we calculate the tangent of the angle θ\theta using the coordinates of u⃗\vec{u}. The tangent of θ\theta is the ratio of the y-coordinate to the x-coordinate.tan⁡(θ)=yx=−8−5=85\tan(\theta) = \frac{y}{x} = \frac{-8}{-5} = \frac{8}{5}
  3. Add 180180 Degrees: Next, we use the arctangent function to find the angle θ\theta whose tangent is 85\frac{8}{5}. However, since the vector is in the third quadrant (both xx and yy are negative), we need to add 180∘180^\circ to the angle we get from the arctangent function to find the correct direction angle that is between 0∘0^\circ and 360∘360^\circ.\newlineθ=atan(85)+180∘\theta = \text{atan}(\frac{8}{5}) + 180^\circ
  4. Calculate Arctangent: We calculate the arctangent of 85\frac{8}{5} using a calculator and then add 180°180° to find the direction angle.\newlineθ≈atan(85)+180°≈58.00°+180°≈238.00°\theta \approx \text{atan}(\frac{8}{5}) + 180° \approx 58.00° + 180° \approx 238.00°
  5. Round to Nearest Hundredth: Finally, we round the direction angle to the nearest hundredth as requested. θ≈238.00∘\theta \approx 238.00^\circ

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