arcos(sinθ)

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November 9th 2010, 02:49 PM
SyNtHeSiS

arcos(sinθ)

If $3 \pi < \theta < \frac{7 \pi}{2}$ , what is $arccos(sin\theta)$ equal to?

Attempt:

Since $3 \pi < \theta < \frac{7 \pi}{2}$ , $\theta = \frac{\pi}{2}$ so:

$arccos(1)$

$= 0$

but think I made a mistake without considering the interval it in. What should I do
November 9th 2010, 03:30 PM
Archie Meade

Quote:

Originally Posted by SyNtHeSiS

If $3 \pi < \theta < \frac{7 \pi}{2}$ , what is $arccos(sin\theta)$ equal to?

Attempt:

Since $3\pi < \theta < \frac{7 \pi}{2}$ , $\theta = \frac{\pi}{2}$ so:

$arccos(1)$

$= 0$

but think I made a mistake without considering the interval it in. What should I do

$3{\pi}\rightarrow\ 3.5{\pi}$ is the third quadrant, $2{\pi}+{\pi}\rightarrow\ 2{\pi}+{\pi}+\frac{\pi}{2}$

$360^o+180^o\rightarrow\ 360^o+180^o+90^o$

If you draw a unit-radius circle, then in that 3rd quadrant, pick a point on the circumference.
draw a right-angled triangle by drawing lines from the point directly up to the x-axis
and directly from the point to the circle centre.

For this triangle, $\alpha=$ angle at the centre inside the triangle.

adjacent $=|x|$

opposite $=|y|$

hypotenuse $=1$

$sin\theta=sin\alpha=-y$

Draw a second right-angled triangle, whose base is $|y|$ and whose opposite is $|x|$ in that same 3rd quadrant

$arccos(-y)=3{\pi}+\frac{\pi}{2}-\alpha$

which is the angle after one revolution in the 3rd quadrant.
There is another angle in the 2nd quadrant, as cosine is negative there also.
From there, the range of angles can be deduced from $0\rightarrow\ 2{\pi}$
November 9th 2010, 04:28 PM
skeeter

Quote:

Originally Posted by SyNtHeSiS

If $3 \pi < \theta < \frac{7 \pi}{2}$ , what is $arccos(sin\theta)$ equal to?

you can only determine the range of values for $\arccos(\sin{\theta})$ .

if $\displaystyle 3 \pi < \theta < \frac{7 \pi}{2}$ , then $-1 < \sin{\theta} < 0$ . Subsequently ...

$\displaystyle \frac{\pi}{2} < \arccos(\sin{\theta}) < \pi$ .
November 9th 2010, 11:19 PM
SyNtHeSiS

Quote:

Originally Posted by Archie Meade

Draw a second right-angled triangle, whose base is $|y|$ and whose opposite is $|x|$ in that same 3rd quadrant

$arccos(-y)=3{\pi}+\frac{\pi}{2}-\alpha$

Why did you swap the x and y around on the triangle when working out the arccos? Also can you say $arccos(-y) = \pi - \beta$ , since its in the 3rd quadrant?
November 10th 2010, 02:08 PM
Archie Meade

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Quote:

Originally Posted by SyNtHeSiS

If $3 \pi < \theta < \frac{7 \pi}{2}$ , what is $arccos(sin\theta)$ equal to?

Attempt:

Since $3 \pi < \theta < \frac{7 \pi}{2},\;\;range\;\;of\;\;\theta = \frac{\pi}{2}$ so:

$arccos(1)$

$= 0$

but think I made a mistake without considering the interval it in. What should I do

Better to start from scratch,

we can ascertain the graph of $acos(sinx)$

$u=sinx$

Notice that $\frac{d}{dx}acos(sinx)=\frac{d}{dx}acos(u)=\frac{d }{du}acos(u)\frac{d}{dx}sinx$

$\displaystyle\ =-\frac{1}{\sqrt{1-u^2}}cosx=-\frac{1}{\sqrt{1-sin^2x}}cosx=-\frac{cosx}{|cosx|}$

This means the slope of the graph is $\pm1$ , depending on whether $cosx$ is positive or negative.

Hence, it is a periodic triangle wave.

For $3{\pi}<\theta<3.5{\pi}$

$\displaystyle\ sin\theta$ goes from $0\rightarrow-1$

Therefore $acos(sin\theta)$ goes from $acos(0)\rightarrow\ acos(-1)$ which is from $\frac{\pi}{2}\rightarrow\ {\pi}$

If you have an exact value for $\theta$ , then since the graph is rising linearly from $3{\pi}\rightarrow\ 3.5{\pi}$ , then

$acos\left(sin\theta}\right)=\frac{\pi}{2}+\theta-3{\pi}$