Determine the sign of sin five pi divided by four without using a calculator.

- anonymous

Determine the sign of sin five pi divided by four without using a calculator.

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- welshfella

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- anonymous

Can you explain to me how to find that? @welshfella :)

- welshfella

what is the sign of opposite / hypotenuse from the diagram

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## More answers

- welshfella

positive angles go anticlockwise
one half a rev is pi and 5pi/4 is another p/4
the signs of the lines are marked

- welshfella

sin 5pi/4 = - / + = -

- anonymous

Both hypotenuse and opposite are negative.

- welshfella

No hypotenuse is always positive

- anonymous

Oops! I was looking at the wrong part. That was the adj.

- welshfella

yes the tangent is positive -/- in 3rd quadrant

- anonymous

"positive angles go anticlockwise" what do you mean by this?

- anonymous

So the answer would be negative, right? But can you explain to me the steps how to find it? I'm so confused right now. Like what do I need to do first?

- welshfella

they are measured in an anticlockwise direction
angles measured in clockwise direction are
negative - that the convention
5pi/4 is equivalent to -3pi/4

- anonymous

\[\sin \frac{ 5 \pi }{ 4 }=\sin \left( \pi+\frac{ \pi }{ 4 } \right)=-\sin \frac{ \pi }{ 4 }=-\frac{ 1 }{ \sqrt{2} }\]

- welshfella

well i've done it using the unit circle . You can also use identities to do these just as surjithayer has just done.

- welshfella

the 1 / sqrt2 comes from the standard triangle
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- anonymous

@surjithayer How did you get \[-\sin \pi/4?\]

- welshfella

sin pi/4 = 1/sqrt2
therefor
- sin pi/4 = -1/sqrt2

- anonymous

\[\sin \left( \pi+\beta \right)=-\sin \beta \]
sin is negative in 3rd quadrant.

- anonymous

Thanks for the help! @surjithayer @welshfella . I'm still kinda confused, but I think I get it now. :)

- welshfella

yw

- anonymous

\[\sin \left( \pi+x \right)=\sin \pi \cos x+\cos \pi \sin x\]
\[\sin \pi=0,\cos \pi=-1\]

- welshfella

its just a personal thing but I find it easier to use diagrams for this type of problem

- anonymous

@welshfella maybe to visualize the problem? Both ways suit for me though. :)

- welshfella

that s good

- anonymous

If I change sin five pi divided by four to, let's say, \[\tan \frac{5\pi }{ 4 }\], what will be the answer? I just want to practice haha.

- anonymous

Will it be positive?

- anonymous

yes it is in 3rd quadrant

- anonymous

Okay, so can I also assume that cos 5pi/4 is negative based on the All Students Take Calculus?

- anonymous

yes

- anonymous

in 3rd quadrant only tan and cot are positive.

- anonymous

Yes, since cot is just the reciprocal of tan. :)

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