$\vec{u}=(-10,-9)$ $\newline$ 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}=(-10,-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}

Calculate Ratio Arctangent: To find the direction angle of the vector $\vec{u} = (-10, -9)$ , 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 gives us the angle whose tangent is the ratio of the y-coordinate to the x-coordinate of the vector.
Convert Radians to Degrees: First, we calculate the arctangent of the ratio of the y-coordinate to the x-coordinate of the vector $\vec{u}$ . Since $\vec{u} = (-10, -9)$ , the ratio is $\frac{-9}{-10}$ . Using a calculator, we find that $\arctan\left(\frac{-9}{-10}\right)$ gives us an angle in radians.
Adjust for Quadrant: We convert the angle from radians to degrees because the question asks for the answer in degrees. To convert radians to degrees, we multiply by $\frac{180}{\pi}$ .
Perform Calculation: The calculated angle will give us the angle relative to the positive x-axis, but since both the x and y coordinates of $\vec{u}$ are negative, $\vec{u}$ lies in the third quadrant. The arctangent function will give us an angle in the first quadrant, so we need to add $180$ ° to get the correct direction angle in the third quadrant.
Add $180$ Degrees: Performing the calculation, we have $\theta = \arctan\left(\frac{-9}{-10}\right) \times \frac{180}{\pi} + 180°$ . Using a calculator, we find that $\theta \approx 42.01° + 180°$ .
Round to Nearest Hundredth: Adding $180$ ° to $42$ . $01$ °, we get $\theta \approx 222.01°$ . This is the direction angle of vector $\vec{u}$ in the third quadrant, measured counterclockwise from the positive x-axis.
Round to Nearest Hundredth: Adding $180$ ° to $42$ . $01$ °, we get $\theta \approx 222.01°$ . This is the direction angle of vector $\vec{u}$ in the third quadrant, measured counterclockwise from the positive x-axis.We round the direction angle to the nearest hundredth as the question asks. Therefore, the direction angle of vector $\vec{u}$ is approximately $222$ . $01$ °.