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Resource: Sine and cosine

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
AngleCosineSine
$$0^\circ=0$$$$1$$$$0$$
$$30^\circ=\frac \pi 6$$$$\frac{\sqrt3}2$$$$\frac12$$
$$45^\circ=\frac \pi 4$$$$\frac{\sqrt2}2$$$$\frac{\sqrt2}2$$
$$60^\circ=\frac \pi 3$$$$\frac12$$$$\frac{\sqrt3}2$$
$$90^\circ=\frac \pi 2$$$$0$$$$1$$
Sine & cosine in 2nd quadrant
AngleCosineSine
$$120^\circ=\frac{2\pi}3$$$$-\frac12$$$$\frac{\sqrt3}2$$
$$135^\circ=\frac{3\pi}4$$$$-\frac{\sqrt2}2$$$$\frac{\sqrt2}2$$
$$150^\circ=\frac{5\pi}6$$$$-\frac{\sqrt3}2$$$$\frac12$$
$$180^\circ=\pi$$$$-1$$$$0$$

Sine & cosine in 3rd quadrant
AngleCosineSine
$$120^\circ=\frac{2\pi}3$$$$-\frac12$$$$\frac{\sqrt3}2$$
$$-150^\circ=-\frac{5\pi}6$$$$-\frac{\sqrt3}2$$$$-\frac12$$
$$-135^\circ=-\frac{3\pi}4$$$$-\frac{\sqrt2}2$$$$-\frac{\sqrt2}2$$
$$-120^\circ=-\frac{2\pi}3$$$$-\frac12$$$$-\frac{\sqrt3}2$$
$$-90^\circ=-\frac \pi 2$$$$0$$$$-1$$
Sine & cosine in 4th quadrant
AngleCosineSine
$$-60^\circ=-\frac \pi 3$$$$\frac12$$$$-\frac{\sqrt3}2$$
$$-45^\circ=-\frac \pi 4$$$$\frac{\sqrt2}2$$$$-\frac{\sqrt2}2$$
$$-30^\circ=-\frac \pi 6$$$$\frac{\sqrt3}2$$$$-\frac12$$

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

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