Exact Trigonometric Values

$\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\qquad \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},\qquad \tan\theta=\frac{\text{opposite}}{\text{adjacent}}.$

For certain angles, these ratios have simple exact forms. For example, $\frac{\sqrt3}{2}$ gives a value precisely; a decimal such as $0.866\ldots$ only approximates it. OCR requires the exact sine and cosine values at $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$ and $90^\circ$, and the exact tangent values at $0^\circ$, $30^\circ$, $45^\circ$ and $60^\circ$.

Build the middle values from two triangles

Imagine an isosceles right-angled triangle whose two shorter sides are each 1 unit. Pythagoras’ theorem gives its hypotenuse as $\sqrt{1^2+1^2}=\sqrt2$ units. Both acute angles are $45^\circ$, so their opposite and adjacent sides are equal. Thus $\sin45^\circ=\cos45^\circ=\frac1{\sqrt2}=\frac{\sqrt2}{2}$, while $\tan45^\circ=1$.

For $30^\circ$ and $60^\circ$, start with an equilateral triangle of side 2 units and draw a perpendicular from its top vertex to the base. This splits it into two right-angled triangles. Each has hypotenuse 2, a base of 1, and a height of $\sqrt{2^2-1^2}=\sqrt3$. Relative to the $30^\circ$ angle, the opposite side is 1 and the adjacent side is $\sqrt3$: $\sin30^\circ=\frac12$, $\cos30^\circ=\frac{\sqrt3}{2}$ and $\tan30^\circ=\frac1{\sqrt3}=\frac{\sqrt3}{3}$. At $60^\circ$, the opposite and adjacent sides swap, giving $\sin60^\circ=\frac{\sqrt3}{2}$, $\cos60^\circ=\frac12$ and $\tan60^\circ=\sqrt3$.

The complete set to know

At the endpoints, a direction along the horizontal has sine 0 and cosine 1; a direction straight up has sine 1 and cosine 0. Tangent is sine divided by cosine, so $\tan0^\circ=0$. At $90^\circ$, that division would have a zero denominator, so tangent is undefined; OCR does not list it among the required tangent values.

Notice that the sine row runs from 0 to 1, while the cosine row contains the same values in reverse order. The two special triangles explain the middle three columns; they also give you a way to reconstruct a value if you forget it.