$\vec{u}=(-7,2)$ $\newline$ 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}=(-7,2)

\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 Tangent Ratio: To find the direction angle of the vector $\vec{u} = (-7, 2)$ , we need to calculate the angle $\theta$ that the vector makes with the positive x-axis. The direction angle can be found using the arctangent function, which is the inverse of the tangent function. The tangent of the direction angle is the ratio of the y-coordinate to the x-coordinate of the vector. However, since the x-coordinate is negative and the y-coordinate is positive, the vector lies in the second quadrant. The arctangent function will give us an angle in the fourth quadrant, so we need to add $180$ degrees to get the angle in the second quadrant.
Find Arctangent Angle: First, calculate the tangent of the direction angle using the y-coordinate and the x-coordinate of the vector $\vec{u}$ : $\newline$ $\tan(\theta) = \frac{y}{x} = \frac{2}{-7}$ .
Adjust for Quadrant: Next, find the arctangent of the ratio to get the angle in degrees. Since most calculators return the angle in the range from $-90$ to $90$ degrees, we will get an angle in the fourth quadrant: $\newline$ $\theta = \arctan\left(\frac{2}{-7}\right)$ .
Calculate Final Angle: Using a calculator, we find that: $\newline$ $\theta \approx \arctan\left(\frac{2}{-7}\right) \approx -15.945^\circ$ . $\newline$ Since this angle is in the fourth quadrant, we add $180$ degrees to find the angle in the second quadrant: $\newline$ $\theta_{\text{second quadrant}} = -15.945^\circ + 180^\circ$ .
Calculate Final Angle: Using a calculator, we find that: $\newline$ $\theta \approx \arctan\left(\frac{2}{-7}\right) \approx -15.945^\circ$ . $\newline$ Since this angle is in the fourth quadrant, we add $180$ degrees to find the angle in the second quadrant: $\newline$ $\theta_{\text{second quadrant}} = -15.945^\circ + 180^\circ$ .Performing the addition, we get: $\newline$ $\theta_{\text{second quadrant}} = 164.055^\circ$ . $\newline$ Now we round this to the nearest hundredth: $\newline$ $\theta_{\text{second quadrant}} \approx 164.06^\circ$ .