how do you find an exact value for sin of 40 degrees without using a calculator

Question

chrisplusian · Answer

the problem says find the exact value of : sin(40)+sin(130)+sin(220)+sin(310) all are in degrees and you cant use a calculator

anonymous · Answer

oh...that makes sense

chrisplusian · Answer

so can you help?

callisto · Answer

For $0 \le x \le 90$
sin x = sinx 
sin (180-x) = sinx 
sin (180+x) = -sinx 
sin(360 - x) = -sinx

In your question, 
sin(130)
= sin (180 - 50)
= sin 50

Use the facts listed in the beginning of the post, do the same for the last two terms. You'll see something nice. Can you do it?

chrisplusian · Answer

what are the properties from the beginning called so that I can look them up in my text book?

chrisplusian · Answer

I am still lost

callisto · Answer

Hmm... I haven't learnt the exact name of these identiteies. Perhaps you can look at the chapter of trigonometry?

chrisplusian · Answer

I am in trig class and the only examples the book uses are the ones involving the standard radian measures. so this is the first one they asked for an exact value that is not a standard radian measure but they dont give an example of how to do it

callisto · Answer

Do you have to convert the angles into radian first?

chrisplusian · Answer

Not in the other problems in this same section

callisto · Answer

Then I assume you don't have to convert it. Is that okay?

chrisplusian · Answer

sure

callisto · Answer

Have you learnt the following?
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chrisplusian · Answer

yes

chrisplusian · Answer

so sine is positive in quadrants one and two

callisto · Answer

Yes.

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callisto · Answer

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sinx = sin (180 - θ) = sin θ 
Got it?

callisto · Answer

Note that in the figure, θ + x = 180