$\vec{u}=(-8,-9)$ $\newline$ Find the direction angle of $\vec{u}$ . Enter your answer as an angle in degrees between $0^{\circ}$ and $360^{\circ}$ rounded to the nearest hundredth. $\newline$ $\theta= \square^{\circ}$

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Algebra 2

Inverses of sin, cos, and tan: degrees

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\vec{u}=(-8,-9)

\newline

Find the direction angle of

\vec{u}

. Enter your answer as an angle in degrees between

0^{\circ}

and

360^{\circ}

rounded to the nearest hundredth.

\newline

\theta= \square^{\circ}

Find direction angle of vector: To find the direction angle of the vector $\vec{u} = (-8, -9)$ , we need to calculate the angle that this vector makes with the positive x-axis. The direction angle $\theta$ can be found using the arctangent function (also known as the inverse tangent or atan), which is the ratio of the y-coordinate to the x-coordinate of the vector.
Calculate arctangent of ratio: First, we calculate the arctangent of the ratio of the y-coordinate to the x-coordinate of the vector $\vec{u}$ . Since $\vec{u} = (-8, -9)$ , we have: $\newline$ $\theta = \arctan\left(\frac{-9}{-8}\right)$ .
Use calculator to find arctan: Using a calculator, we find that: $\newline$ $\theta = \arctan\left(\frac{-9}{-8}\right) \approx \arctan(1.125)$ .
Adjust angle for correct quadrant: The $\text{arctan}$ of $1.125$ gives us an angle in the first quadrant, but since both the $x$ and $y$ components of the vector are negative, the vector is actually in the third quadrant. We need to add $180^\circ$ to the angle we get from the $\text{arctan}$ function to place it in the correct quadrant.
Calculate final angle: Calculating the angle, we get: $\newline$ $\theta \approx \arctan(1.125) + 180°$ .
Round angle to nearest hundredth: Using a calculator, we find that: $\newline$ $\theta \approx 48.37° + 180°$ .
Round angle to nearest hundredth: Using a calculator, we find that: $\newline$ $\theta \approx 48.37° + 180°$ .Adding $180$ ° to $48$ . $37$ °, we get: $\newline$ $\theta \approx 228.37°$ .
Round angle to nearest hundredth: Using a calculator, we find that: $\newline$ $\theta \approx 48.37° + 180°$ .Adding $180$ ° to $48$ . $37$ °, we get: $\newline$ $\theta \approx 228.37°$ .We round the angle to the nearest hundredth, as requested: $\newline$ $\theta \approx 228.37°$ .