Silly question about trig past pi/2

[motornoob101]

I know all the well know angles from 0 to $$\pi/2$$. However, past pi/2 I am rather clueless and have no idea what angle corresponds to what value.

For example, in trying to figure out $$tan^{-1}(-1)$$ How does one know it is 5pi/4 and not 7pi/8? I was debating between measuring pi/4 from the negative x-axis (which would give me 7pi/8) and measuring pi/4 from the y axis,(which would give me 5pi/4) but as you can see, I choose the wrong value. Is there a trick to it? Thanks.

 

[sutupidmath]

well

$$arctan(-1)=-arctan(1)=-\frac{\pi}{4}$$ so it means that $$\frac{\pi}{4}$$ lies on the fourth quadrant of the unit circle. So you can get that value by subtracting it from $$2\pi$$ that is

$$2\pi-\frac{\pi}{4}=\frac{8\pi-\pi}{4}=\frac{7\pi}{4}$$ because -pi/4 is the angle counted clockwise direction.
YOu know that it does not lie on the 3rd quadrant because tan(x) there is positive. There are other tricks too.

 

[JohnDuck]

That's the "problem" with inverse trig functions--since, for example, there are an infinite number of solutions for y to the equation $\tan{y} = x$, in order for $f(x) = \arctan{x}$ to be a well-defined function, you have to consider it as mapping to a particular interval of angles. Without qualification, arctan typically returns values between $\frac{-\pi}{2}$ and $\frac{\pi}{2}$, but any interval of length $\pi$ is valid as long as your usage is consistent. There are similar considerations for arcsin and arccos.

Edit: Because I'm not sure if I made it clear, let me say this: $\arctan{x}$ is usually defined as the function from the real numbers to the interval $(\frac{-\pi}{2} , \frac{\pi}{2})$ such that $\tan{(\arctan{x})} = x$ for any real x.

 

[motornoob101]

Oh I forget that arctan is an odd function. Thanks. But what if the question asks for a value for arctan(-1) within pi? I would have tried the 7pi/4 approach if I wasn't restricted to 0 to pi. Thanks.

 

[sutupidmath]

Oh I forget that arctan is an odd function. Thanks. But what if the question asks for a value for arctan(-1) within pi? I would have tried the 7pi/4 approach if I wasn't restricted to 0 to pi. Thanks.

If you have to find the value within an interval of a length $$\pi$$ that doent mean that your interval must be $$[0,\pi]$$ , moreover the common interval for this problem is chosen like JohnDuck stated $$[-\frac{\pi}{2},\frac{\pi}{2}]$$.

This means that also $$\frac{7\pi}{2}$$ is within that interval.

 

[JohnDuck]

One more thing $$arctan(-1)$$ has no value that lies on the interval $$[0,\pi]$$.

$$\frac{3\pi}{4}$$?

 

[sutupidmath]

$$\frac{3\pi}{4}$$?

Blahhhhhh! Yup.