Vector Components and Finding the Correct Angle – Quadrants, Arctangent, and the Unit Circle

Free Preview of The Ultimate Crash Course for STEM Majors

This lesson is nothing more than a free preview of The Ultimate Crash Course for STEM Majors.

Visit AuthorJonD.com to explore the Crash Course series, books, lessons, and additional educational material.

You can also check out my fiction, books, stories, and other creative work at AuthorJonD.com.

For example, the components of $\vec{A}$

If at the origin, the angle starting from 0°

Note* If you position all the tails at the origin, then you can take the angle starting at 0° going to 360° and then using sine for y-axis and cosine for x-axis.

NOTE* Be careful to pay attention to signs—that is, $\left(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right)$ is in Quadrant III but coordinates calculated using $f^{-1}(\theta)=\arctan\theta\equiv\tan^{-1}\theta\neq\frac{1}{\tan\theta}$ [inverse vs. reciprocal].

$$ \tan\theta=\frac{y}{x} \quad\Rightarrow\quad \theta=\arctan\frac{y}{x} \quad\Rightarrow\quad \theta= \arctan \left[ \frac{-\frac{\sqrt{2}}{2}} {-\frac{\sqrt{2}}{2}} \right] = \tan^{-1}(1) \neq 45^\circ, \text{ why?} $$

$\theta=\arctan 1$

The Unit Circle

If theta has a domain ranging from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$

$$ D= \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) = (-90^\circ,90^\circ) $$

Then end points are not included because that would situation a zero in the denominator for x. I.e., $\arctan\frac{y}{0}$ which is undefined.

The correct angle is in QIII which is 225° or 45° + 180°. Found using the unit circle but your angle may not be on the circle, so you’ll have to add 90° or 190° based on QII and III.

The Unit Circle

$$ (1,0)=0^\circ $$

$$ (0,1)=90^\circ=\frac{\pi}{2} $$

$$ (-1,0)=180^\circ=\pi $$

$$ (0,-1)=270^\circ=\frac{3\pi}{2} $$

Quadrant I: $0^\circ<\theta<90^\circ$

Quadrant II: $90^\circ<\theta<180^\circ$

Quadrant III: $180^\circ<\theta<270^\circ$

Quadrant IV: $270^\circ<\theta<360^\circ$

NOTE* The tiny little details are the enemy of all students. You need to seek the tiny details!

Continue The Ultimate Crash Course for STEM Majors

This lesson is nothing more than a free preview of The Ultimate Crash Course for STEM Majors. The complete Crash Course series provides additional worked examples and lessons covering algebra, calculus, physics, vectors, trigonometry, differential equations, and other STEM topics.

Explore The Ultimate Crash Course for STEM Majors at AuthorJonD.com →