6.8 - Exact trigonometric values

The forty-five-degree triangle

Can you work out $\tan30^\circ\times\tan60^\circ$ exactly without knowing either decimal? Two simple triangles will supply everything you need, and we will return to that product at the end.

Start with a right-angled triangle whose two shorter sides both have length $1$. Its two acute angles are equal and add to $90^\circ$, so each is $45^\circ$. Pythagoras gives a hypotenuse of $\sqrt{1^2+1^2}=\sqrt2$.

[DIAGRAM: asset_name: The forty-five-degree triangle; asset_slug: l2fm-exact-45-triangle; caption: Equal legs of length 1 give a hypotenuse of square root 2 and two forty-five-degree angles; description: A right-angled isosceles triangle has horizontal and vertical legs each labelled 1, hypotenuse labelled square root 2 and both acute angles labelled 45 degrees. A right-angle square marks the perpendicular legs. The equation 1 squared plus 1 squared equals square root 2 squared and ratio 1 to 1 to square root 2 are shown beside it.]

Any enlargement keeps the angles, so every $45^\circ,45^\circ,90^\circ$ triangle has the same side ratio. A ratio specifies relative lengths, not a requirement that one actual side must be $1$.

The sheet gives the general right-triangle sine, cosine and tangent ratios. It does not give these exact values. Use the ratio to scale lengths directly.

Worked example 2 changes only the hypotenuse to $10\sqrt2\,\mathrm{cm}$. The same division now produces an integer.

The ratio's order matters: its longest entry must represent the hypotenuse.

Apply the same scaling to a new hypotenuse.

The thirty- and sixty-degree triangle

Bisect an equilateral triangle to obtain a second set of exact ratios.

Take an equilateral triangle of side $2$. Its perpendicular bisector splits the base into two lengths of $1$, and its top angle into two angles of $30^\circ$. Each half is a right-angled triangle with hypotenuse $2$, short leg $1$ and long leg $\sqrt{2^2-1^2}=\sqrt3$.

[DIAGRAM: asset_name: The thirty- and sixty-degree triangle; asset_slug: l2fm-exact-30-60-triangle; caption: Bisecting an equilateral triangle gives the exact side ratio 1 to square root 3 to 2; description: An equilateral triangle of side 2 is split down its perpendicular height into two right-angled triangles. Each half-base is labelled 1, the height is square root 3, an outside sloping side is 2, and the right-hand half has acute angles 30 degrees at the top and 60 degrees at the base. A right-angle square marks the height meeting the base.]

The side opposite $30^\circ$ is the short leg. The side opposite $60^\circ$ is the long leg. Saying which angle a side is opposite avoids swapping them.

Changing which side is given changes which ratio entry you match first.