Sine and cosine are mathematical functions defined
as the horizontal and vertical “length” or coordinate, respectively, to any
point on a unit circle. Different angles give different sine/cosine pairs. Some
may be useful to memorise.

Some useful angles are given in degrees and radians along with their sine and cosine values. Derivations of the values can be found in Proof 21.

Sine & cosine in 1st quadrant |

**Angle** | **Cosine** | **Sine** |

$$0^\circ=0$$ | $$1$$ | $$0$$ |

$$30^\circ=\frac \pi 6$$ | $$\frac{\sqrt3}2$$ | $$\frac12$$ |

$$45^\circ=\frac \pi 4$$ | $$\frac{\sqrt2}2$$ | $$\frac{\sqrt2}2$$ |

$$60^\circ=\frac \pi 3$$ | $$\frac12$$ | $$\frac{\sqrt3}2$$ |

$$90^\circ=\frac \pi 2$$ | $$0$$ | $$1$$ |

Sine & cosine in 2nd quadrant |

**Angle** | **Cosine** | **Sine** |

$$120^\circ=\frac{2\pi}3$$ | $$-\frac12$$ | $$\frac{\sqrt3}2$$ |

$$135^\circ=\frac{3\pi}4$$ | $$-\frac{\sqrt2}2$$ | $$\frac{\sqrt2}2$$ |

$$150^\circ=\frac{5\pi}6$$ | $$-\frac{\sqrt3}2$$ | $$\frac12$$ |

$$180^\circ=\pi$$ | $$-1$$ | $$0$$ |

Sine & cosine in 3rd quadrant |

**Angle** | **Cosine** | **Sine** |

$$120^\circ=\frac{2\pi}3$$ | $$-\frac12$$ | $$\frac{\sqrt3}2$$ |

$$-150^\circ=-\frac{5\pi}6$$ | $$-\frac{\sqrt3}2$$ | $$-\frac12$$ |

$$-135^\circ=-\frac{3\pi}4$$ | $$-\frac{\sqrt2}2$$ | $$-\frac{\sqrt2}2$$ |

$$-120^\circ=-\frac{2\pi}3$$ | $$-\frac12$$ | $$-\frac{\sqrt3}2$$ |

$$-90^\circ=-\frac \pi 2$$ | $$0$$ | $$-1$$ |

Sine & cosine in 4th quadrant |

**Angle** | **Cosine** | **Sine** |

$$-60^\circ=-\frac \pi 3$$ | $$\frac12$$ | $$-\frac{\sqrt3}2$$ |

$$-45^\circ=-\frac \pi 4$$ | $$\frac{\sqrt2}2$$ | $$-\frac{\sqrt2}2$$ |

$$-30^\circ=-\frac \pi 6$$ | $$\frac{\sqrt3}2$$ | $$-\frac12$$ |

Note how it is sufficient to memorise the five values from the 1^{st} quadrant. Those are repeated to all other quadrants due to symmetry, just with an added negative sign here and there.

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