Finding the direction of an angle in the unit circle

[aron silvester]

Homework Statement

I'm having trouble understanding how to find the angle of a vector. Here we are given the x and y component to help us find the direction of vector C. In this case, both x and y component is negative, so it should be in the third quadrant. I know that since we have both the x and y component, we need to use inverse tangent, and since the vector is in the third quadrant we add 180 to the inverse tangent as shown in part 3 because the vector is past 180, but before 270. My question is if the vector is in the second quadrant, would we add 90 to inverse tangent? If the vector is in the fourth quadrant, would we add 270 to inverse tangent? This is seriously frustrating me.

Homework Equations

It's all in part 3.

The Attempt at a Solution

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[TSny]

Most calculators give only the "principal value" of the inverse tangent function. https://en.wikipedia.org/wiki/Inverse_trigonometric_functions#Principal_values

For the first quadrant you are OK.

For the fourth quadrant, you are also OK. Your calculator should give a negative answer, such as -30^o. This is interpreted to mean that the vector is pointing 30^o below the positive x axis. If you want, you can express the angle as 360^o - 30^o = 330^o counterclockwise from the + x axis.

For the second and third quadrants you add 180^o to the calculator's output. The result is then the angle as measured counterclockwise from the positive x axis.

So, if the vector is in the second quadrant, your calculator would give a negative value, say -40^o. Adding 180^o to this gives 140^o. So, the direction of the vector is 140^o counterclockwise from the positive x axis.

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Another approach is to always take the absolute value of the output of the calculator and express the direction as "above the positive x axis", "above the negative x axis", "below the negative x axis", or "below the positive x axis" depending on being in the first, second, third, or fourth quadrant, respectively. You can always tell which quadrant by inspecting the signs of the x and y components of the vector.

In any case, avoid expressing the direction of a vector with just an angle. Always include an explanatory phrase such as "counterclockwise from the positive x axis" or "above the negative x axis", etc. Or, include a diagram showing the direction of the vector with the angle indicated on the diagram.

Note: Some calculators have polar coordinate features which will always give you the answer as measured counterclockwise from the positive x-axis so that you don't need to do anything extra, no matter which quadrant. All you need to do is input the x and y components of the vector, including sign. You don't need to use the inverse tangent function. For an example, see


Of course details will vary with the brand of the calculator.

 

[aron silvester]

Most calculators give only the "principal value" of the inverse tangent function. https://en.wikipedia.org/wiki/Inverse_trigonometric_functions#Principal_values

For the first quadrant you are OK.

For the fourth quadrant, you are also OK. Your calculator should give a negative answer, such as -30^o. This is interpreted to mean that the vector is pointing 30^o below the positive x axis. If you want, you can express the angle as 360^o - 30^o = 330^o counterclockwise from the + x axis.

For the second and third quadrants you add 180^o to the calculator's output. The result is then the angle as measured counterclockwise from the positive x axis.

So, if the vector is in the second quadrant, your calculator would give a negative value, say -40^o. Adding 180^o to this gives 140^o. So, the direction of the vector is 140^o counterclockwise from the positive x axis.

-----------------------------------------

Another approach is to always take the absolute value of the output of the calculator and express the direction as "above the positive x axis", "above the negative x axis", "below the negative x axis", or "below the positive x axis" depending on being in the first, second, third, or fourth quadrant, respectively. You can always tell which quadrant by inspecting the signs of the x and y components of the vector.

In any case, avoid expressing the direction of a vector with just an angle. Always include an explanatory phrase such as "counterclockwise from the positive x axis" or "above the negative x axis", etc. Or, include a diagram showing the direction of the vector with the angle indicated on the diagram.

Note: Some calculators have polar coordinate features which will always give you the answer as measured counterclockwise from the positive x-axis so that you don't need to do anything extra, no matter which quadrant. All you need to do is input the x and y components of the vector, including sign. You don't need to use the inverse tangent function. For an example, see


Of course details will vary with the brand of the calculator.

Thanks!