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Find the direction angle of $\mathbf{u} = (-10,7)$ . Enter your answer as an angle in degrees between $0^\circ$ and $360^\circ$ rounded to the nearest hundredth.

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Inverses of trigonometric functions using a calculator

Full solution

Q. Find the direction angle of

\mathbf{u} = (-10,7)

. Enter your answer as an angle in degrees between

0^\circ

and

360^\circ

rounded to the nearest hundredth.

Definition of Direction Angle: The direction angle of a vector in the plane is the angle measured counterclockwise from the positive $x$ -axis to the vector. To find this angle, we use the arctangent function, which gives us the angle whose tangent is the ratio of the $y$ -coordinate to the $x$ -coordinate of the vector.
Calculate Tangent Ratio: For the vector $\mathbf{u} = (-10,7)$ , the x-coordinate is $-10$ and the y-coordinate is $7$ . The tangent of the direction angle $\theta$ is therefore $\tan(\theta) = \frac{7}{-10} = -0.7$ .
Use Arctangent Function: We use the arctangent function to find the angle $\theta$ such that $\tan(\theta) = -0.7$ . However, since the arctangent function returns values between $-\pi/2$ and $\pi/2$ (or $-90$ degrees and $90$ degrees), and our vector is in the second quadrant (where both sine and cosine are negative), we need to add $180$ degrees to the arctangent value to get the correct direction angle.
Find Angle in Second Quadrant: Calculating the arctangent of $-0.7$ using a calculator, we get $\theta = \text{arctan}(-0.7) \approx -35.00$ degrees. Since the vector is in the second quadrant, we add $180$ degrees to this value to find the actual direction angle.
Calculate Final Direction Angle: Adding $180$ degrees to $-35.00$ degrees, we get the direction angle as $180 - 35.00 = 145.00$ degrees.

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