When theta must be positive and when negative

Aug 2016
74
3
Plymouth, MA
I'm grappling with just not seeing when to use negative angles and when to use positive angles when solving trig functions.

If I pick a point on a graph with coordinates (5,-5) the angle formed between the x-axis and the hypotenuse can be either $-45^\circ$ or $315^\circ$ (along with all multiples of $360^\circ$). Yes?

If this is true, then I would think that $\sin(-45^\circ) = \sin(315^\circ) = \dfrac{-\sqrt{2}}{2}$

If I plot the sine function I see that $\sin(-45^\circ) = \dfrac{-\sqrt{2}}{2}$ but my calculator gives me an error when entering $\sin(-45^\circ)$ or $\sin(\dfrac{-\pi}{4})$

Likewise, my current lesson on complex numbers requires that I use the sin and cos functions of $\dfrac{7\pi}{4}$ rather than $\dfrac{-\pi}{4}$

I would appreciate an explanation of when to use a negative reference angle and when to use the positive angle $< 360^\circ$
 

Plato

MHF Helper
Aug 2006
22,469
8,640
I'm grappling with just not seeing when to use negative angles and when to use positive angles when solving trig functions.

If I pick a point on a graph with coordinates (5,-5) the angle formed between the x-axis and the hypotenuse can be either $-45^\circ$ or $315^\circ$ (along with all multiples of $360^\circ$). Yes?
If this is true, then I would think that $\sin(-45^\circ) = \sin(315^\circ) = \dfrac{-\sqrt{2}}{2}$
If I plot the sine function I see that $\sin(-45^\circ) = \dfrac{-\sqrt{2}}{2}$ but my calculator gives me an error when entering $\sin(-45^\circ)$ or $\sin(\dfrac{-\pi}{4})$
Likewise, my current lesson on complex numbers requires that I use the sin and cos functions of $\dfrac{7\pi}{4}$ rather than $\dfrac{-\pi}{4}$
I would appreciate an explanation of when to use a negative reference angle and when to use the positive angle $< 360^\circ$
Some older texts & authors still use degrees as well as reference angles which acute. If the terminal side is in quadrant II the reference angle is $180^{\circ}-\theta$;
terminal side is in quadrant III the reference angle is $180^{\circ}+\theta$;terminal side is in quadrant IV the reference angle is $360^{\circ}-\theta$.
In II the $\sin$ is positive, in III the $\tan$ is positive, & in IV the $cos$ is positive.

Look Here. The basic parts of wolframalpha are free to use. The whole site is inexpensive for students & the is a version for small screen devices.
 
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ChipB

MHF Helper
Jun 2014
305
124
NJ
I don't know why your calculator can't determine sin(-pi/4)). Mine is ok with it, as well as sin(7 pi/4)

Whether to measure angles as clockwise from the x-axis or counterclockwise depends on the application, and may be influenced by the particulars of the problem you are working. As an example, to calculate the y-coordinate of a point on a wheel that is rotating about the origin you would use y=R sin(theta) where theta is positive if the wheel rotates counterclockwise and theta is negative if it's rotating clockwise. This allows for a nice smooth plot of theta and y versus time. As for complex analysis - you are correct that by convention angles are measured clockwise from the real axis, and of course they are given in radians.
 
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Aug 2016
74
3
Plymouth, MA
Really, thanks ChipB.

As for the calculator, it was user error. Much to my chagrin, I keep forgetting the TI-84 requires the use of the (-) key rather than the minus key for negative values.

My Trig and Pre-Calc course is a self-study program. I worry sometimes when I don't come up with the same answers as the text.

As an example:

Find Trigonometric form of $5 - 5i$. I came up with $z = 5\sqrt{2} (\cos\dfrac{-\pi}{4}+i \sin\dfrac{-\pi}{4})$

But the text answer was $z = 5\sqrt{2} (\cos\dfrac{7\pi}{4}+i \sin\dfrac{7\pi}{4})$

From your answer it seems either should be accepted as correct. Is that true? Would there normally be a preference?
 

Plato

MHF Helper
Aug 2006
22,469
8,640
Really, thanks ChipB.
As for the calculator, it was user error. Much to my chagrin, I keep forgetting the TI-84 requires the use of the (-) key rather than the minus key for negative values.
My Trig and Pre-Calc course is a self-study program. I worry sometimes when I don't come up with the same answers as the text.
As an example:
Find Trigonometric form of $5 - 5i$. I came up with $z = 5\sqrt{2} (\cos\dfrac{-\pi}{4}+i \sin\dfrac{-\pi}{4})$
But the text answer was $z = 5\sqrt{2} (\cos\dfrac{7\pi}{4}+i \sin\dfrac{7\pi}{4})$
From your answer it seems either should be accepted as correct. Is that true? Would there normally be a preference?
The "angle" associated with a complex is known as its argument. So \(\displaystyle \arg (5 - 5\bf{i}) = - \frac{\pi }{4}\)
Note that I used the negative value. In resent times that has become the preferred notation. However older texts often used the convention that
\(\displaystyle 0\le \arg (z) <2\pi\) that seems to be the case with your textbook.
Just note that $2\pi -\frac{\pi}{4}=\frac{7\pi}{4}$ That means that the two are equivalent.
 
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Aug 2016
74
3
Plymouth, MA
Thanks for the confirmation, Plato.