# Why is $\sin(155^\circ)$ same as $\sin(25^\circ)$?

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?

• More generally, $\sin(180^\circ - x) = \sin(x)$ Can you think of the sine in terms of right triangles? – GEdgar Jun 27 '18 at 0:11
• Related (possibly duplicate): "How to remember a particular class of trig identities". The titular class includes $\sin(\pi -x)= \sin x$. – Blue Jun 27 '18 at 1:29

## 2 Answers

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.

• Thank you :D now i see... But shouldn't one of them give -0.42 ? instead of both giving +0.42? – Feelsbadman Jun 27 '18 at 0:18
• they are both the same height. To get -0.42 ad 180 degrees so sin(205). – Chris2018 Jun 27 '18 at 0:21
• 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} – Michael Hardy Jun 27 '18 at 0:36

$$\sin ( \pi - \alpha ) = \sin \pi \cos \alpha -\cos \pi \sin \alpha = \sin \alpha$$