### Trigonometry: Angles with sine, cosine and tangent

### Symmetry in the unit circle

We have already *seen* that the sine and cosine repeat themselves every #2 \pi#. Now we will look at symmetry in the unit circle.

We can find three kinds of symmetry in the unit circle, namely reflection across the #x#-axis, reflection across the #y#-axis and reflection across the line #y=x#. This symmetry gives us the following rules for sine and cosine.

Reflection across the |
Sine |
Cosine |

#x#-axis |
#\sin(-\alpha) = -\sin(\alpha)# |
#\cos(-\alpha) = \cos(\alpha)# |

#y#-axis |
#\sin(\pi-\alpha) = \sin(\alpha)# |
#\cos(\pi-\alpha) = -\cos(\alpha)# |

line #y=x# |
#\sin(\frac{\pi}{2}-\alpha) = \cos(\alpha)# |
#\cos(\frac{\pi}{2}-\alpha) = \sin(\alpha)# |

Through these rules, we only need to know the values of the sine and cosine in the first quarter of the unit circle, i.e. the values of sine and cosine for #0 \leq \alpha \leq \tfrac{\pi}{4}#. In practice, we take a slightly larger piece and look specifically at the values of the #0 \leq \alpha \leq \tfrac{\pi}{2}#.

Given is #\sin\left(\frac{\pi}{4}\right)=\dfrac{1}{\sqrt{2}}#. What is #\sin\left(\frac{3 \pi}{4}\right)#?

The given angles are related to each other through reflection across the #y#-axis. Hence \[\sin\left(\frac{3 \pi}{4}\right)=\sin\left(\frac{\pi}{4}\right)=\dfrac{1}{\sqrt{2}}\]

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