Euler's Eyes
2026-10-21
Demonstrations of the values of 30, 45 and 60 degrees
Two drawings replace memorization
The trigonometric ratios of an acute angle are:
$$\sin\alpha=\frac{\text{cateto oposto}}{\text{hipotenusa}},\quad \cos\alpha=\frac{\text{cateto adjacente}}{\text{hipotenusa}},$$$$\tan\alpha=\frac{\text{cateto oposto}}{\text{cateto adjacente}}.$$
They can be calculated exactly for some angles using Pythagoras. See the metric relations class in this track for the proof of the theorem.
The 45 degree triangle
The diagonal of a square with side 1 divides it into two isosceles right triangles. The acute angles are equal and add up to $90°$; each measures $45°$.
According to Pythagoras’ theorem, the diagonal measures $\sqrt{1^2+1^2}=\sqrt2$. Logo:
$$\sin45°=\cos45°=\frac1{\sqrt2}=\frac{\sqrt2}{2},\qquad\tan45°=1.$$The triangle of 30 and 60 degrees
Plot the height of one side equilateral triangle 2. It divides the base into two parts of length 1 and the vertex angle into two parts of $30°$.
Height $h$ satisfies $h^2+1^2=2^2$, then $h=\sqrt3$. Reading opposite and adjacent legs:
| Angle | Sine | Cosine | Tangent |
|---|---|---|---|
| $30°$ | $1/2$ | $\sqrt3/2$ | $\sqrt3/3$ |
| $45°$ | $\sqrt2/2$ | $\sqrt2/2$ | $1$ |
| $60°$ | $\sqrt3/2$ | $1/2$ | $\sqrt3$ |
In $30°$ and $60°$, the cats switch roles. This explains why the sine of one coincides with the cosine of the other.
Exact values and approximations
$\sqrt3/2$ is an exact value. The decimal $0{,}866$ is an approximation. Keep the radical throughout the calculation and round only when it is necessary to interpret a measurement.
In everyday life
- Squares. School squares have the angles of $30^\circ$, $45^\circ$, $60^\circ$ and $90^\circ$.
- Roofs. Roofs of $30^\circ$ and $45^\circ$ are common: with a half-width of 4 m, a roof of $45^\circ$ rises 4 m.
- Honeycombs. Bees make hexagons, whose inner triangles are equilateral, with angles of $60^\circ$.
Exercises with resolution
1. Calculate $\sin30°+\cos60°+\tan45°$.
Check resolution 1
$1/2+1/2+1=2$.
2. One equilateral triangle has side 12 cm. Calculate your height.
Check resolution 2
The height is $12\sin60°=6\sqrt3$ cm. It can also be obtained by $h^2+6^2=12^2$.
In the interactive book
The same subjects, with all the demonstrations and the complete list of exercises, are in the interactive Trigonometry book:
This class is a shorter introduction to the topic; In the book, the theory is accompanied by solved and proposed exercises from the chapter.