$\vec{u}=(-7,-10)$ $\newline$ 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}=(-7,-10)

\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 direction angle: To find the direction angle of the vector $\vec{u} = (-7, -10)$ , we need to calculate the angle that this vector makes with the positive x-axis. The direction angle, often denoted as $\theta$ , can be found using the arctangent function ( $\tan^{-1}$ or $\text{atan}$ ), which gives us the angle whose tangent is the ratio of the y-coordinate to the x-coordinate of the vector.
Find tangent of angle: First, we calculate the tangent of the angle $\theta$ using the y-coordinate and the x-coordinate of the vector $\vec{u}$ . The tangent of $\theta$ is given by $\tan(\theta) = \frac{y}{x}$ . For $\vec{u} = (-7, -10)$ , this is $\tan(\theta) = \frac{-10}{-7}$ .
Take arctangent: Performing the division, we get $\tan(\theta) = \frac{10}{7}$ . Now we need to take the arctangent of this value to find the angle $\theta$ in radians. However, since we want the angle in degrees and between $0^\circ$ and $360^\circ$ , we will need to adjust the angle we get from the arctangent function accordingly.
Convert to degrees: Using a calculator, we find the arctangent of $\frac{10}{7}$ , which gives us $\theta$ in radians. To convert this to degrees, we multiply by $\frac{180}{\pi}$ . The calculator will give us an angle in the first quadrant, but since the vector has negative $x$ and $y$ components, the actual direction angle is in the third quadrant.
Adjust for quadrant: The angle from the arctangent function is approximately \atan(\frac{10}{7}) \approx 55.00^\circ. Since the vector is in the third quadrant, we add $180^\circ$ to this angle to find the correct direction angle $\theta$ .
Final direction angle: Adding $180^\circ$ to $55.00^\circ$ gives us $\theta \approx 235.00^\circ$ . This is the direction angle of the vector $\vec{u}$ in the third quadrant, which is between $0^\circ$ and $360^\circ$ .