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I am practicing for a test and i've come across this question which asks "What is the value of $\sin(25^\circ)$ if $\sin(155^\circ) = 0.423$?"

and I've checked on the calculator, both give same result; $0.423.$ why do they have the same value? how would you know this without using a calculator?

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    $\begingroup$ More generally, $\sin(180^\circ - x) = \sin(x)$ Can you think of the sine in terms of right triangles? $\endgroup$ – GEdgar Jun 27 '18 at 0:11
  • $\begingroup$ Related (possibly duplicate): "How to remember a particular class of trig identities". The titular class includes $\sin(\pi -x)= \sin x$. $\endgroup$ – Blue Jun 27 '18 at 1:29
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try drawing a circle sin is the height. You will see that at the angle 155 degrees it is the same height as 25 degrees.

enter image description here

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  • $\begingroup$ Thank you :D now i see... But shouldn't one of them give -0.42 ? instead of both giving +0.42? $\endgroup$ – Feelsbadman Jun 27 '18 at 0:18
  • $\begingroup$ they are both the same height. To get -0.42 ad 180 degrees so sin(205). $\endgroup$ – Chris2018 Jun 27 '18 at 0:21
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    $\begingroup$ I upvoted this answer. But this illustration reminds me of those images accompanying some Wikipedia articles where you can't edit them to clean up the typesetting, so you see things like $$ \begin{align} & \sin\,\alpha{=}\sin\,(\pi\text{-}\alpha) & \text{instead of} & \qquad \sin\alpha = \sin(\pi-\alpha) \\ & \cos\,(\pi\text{-}\alpha) = \text{-}\,\cos\,\alpha & \text{instead of} & \qquad \cos(\pi-\alpha) = -\cos\alpha \end{align} $$ $\endgroup$ – Michael Hardy Jun 27 '18 at 0:36
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$$ \sin ( \pi - \alpha ) = \sin \pi \cos \alpha -\cos \pi \sin \alpha = \sin \alpha $$

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