Trigonometry

Radians, arc length and trigonometric cycle

Compare an arc with its radius If an arc has length $s$ in a circle of radius $R>0$, its measure in radians is: $$\theta=\frac{s}{R}.$$A radian is the central angle that cuts an arc as long as the radius. Since the length of a circle is $2\pi R$, one revolution measures $2\pi$ radians. It follows that: $$360°=2\pi\text{ rad},\qquad180°=\pi\text{ rad}.$$The conversions are $\theta_{\mathrm{rad}}=\theta_{\mathrm{graus}}\pi/180$ and $\theta_{\mathrm{graus}}=\theta_{\mathrm{rad}}180/\pi$. For example, $75°=5\pi/12$ rad. Demonstrate the length formula An arc of $\alpha$ degrees occupies fraction $\alpha/360$ of the turn. Its length is:

October 20, 2026 Read

Triangulation: measuring a distance without crossing

Imagine a ball sitting on the other side of a stream. You want to know the distance to it, but you don’t You cannot cross or stretch a tape measure across the water. A triangle can help: instead of measure the entire path, we measure a safe section and observe two angles. A triangle that reveals a measurement Let’s call P the point we want to reach. On the margin, find a point A so that the segment from A to P forms a right angle with the margin. Mark another point B in the margin, the some distance from A.

October 20, 2026 Read

Any triangles: the complete toolbox

In the interactive book: complete chapter X, with all demonstrations and exercises. A bridge seen from the bank Two points A and B are on the same bank of a river, separated by a known distance. A point C is on the other bank. By measuring the angles at A and B, we can calculate AC and BC without crossing. The triangle formed may not have any right angles. Let’s call the angles $A,B,C$ and the opposite sides $a,b,c$. Always $A+B+C=180°$. The sides of a nondegenerate triangle are positive and each is less than the sum of the other two.

October 20, 2026 Read

Right triangles: measuring without reaching

In the interactive book: complete chapter IX, with all demonstrations and exercises. The height of the tree You are 6 m from a tree, on horizontal ground, and you see its top with an elevation angle of $45°$. It is possible to find out the height without climbing it. We need to set up a triangle and say exactly where we observed it from. A right triangle has an angle of $90°$. The side opposite to it is the hypotenuse; the others are the cathetes. In relation to a chosen acute angle, one side is opposite and the other adjacent.

October 20, 2026 Read

Inequalities: choosing pieces of the loop

In the interactive book: complete chapter VIII, with all demonstrations and exercises. Higher than a line Question $\operatorname{sen} x>1/2$ asks for all positions where the cab is above height line $1/2$. In one turn, the cab crosses this line at $30°$ and $150°$. Between them, it passes through $90°$, where the height is 1. Therefore, in $[0°,360°)$: $$30°

October 20, 2026 Read

Equations: looking for hidden angles

In the interactive book: complete chapter VII, with all demonstrations and exercises. When does the cabin reach the same height? Looking for $\operatorname{sen} x=1/2$ is asking in which positions the cabin is at height $1/2$. A horizontal line cuts the cycle at two points: one in the first quadrant and one in the second. In one turn, these are $30°$ and $150°$. In this chapter, the main examples use degrees. When $\pi$ appears, the angle will be in radians.

October 20, 2026 Read

Transformations: building new formulas

In the interactive book: complete chapter VI, with all demonstrations and exercises. Adding turns is not adding heights A turn of $30°$ followed by another of $30°$ results in $60°$. But $\operatorname{sen}30°+\operatorname{sen}30°=1$, while $\operatorname{sen}60°=\sqrt3/2$. We need to find out how the coordinates change when rotating. This is an advanced step. If the letter beads are still new, follow the example of $75°$ first and return to the demonstration with an adult.

October 20, 2026 Read

Notable Angles: Folding Triangles and Polygons

In the interactive book: complete chapter V, with all demonstrations and exercises. Make a square Draw a square with side 1 and draw a diagonal. It divides two right angles in half: two triangles with angles of $45°$, $45°$ and $90°$ appear. According to Pythagoras’ theorem, the diagonal measures $\sqrt{1^2+1^2}=\sqrt2$. For an acute angle in a right triangle, sine is the opposite side divided by the hypotenuse; cosine is the adjacent side divided by the hypotenuse. These ratios coincide with the cycle coordinates: just reduce the hypotenuse to the measure 1, without changing the angles.

October 19, 2026 Read

Symmetries: one table serves the entire round

In the interactive book: complete chapter IV, with all demonstrations and exercises. The wheel mirror Draw a point in the first quadrant and its reflection in the second. The height is the same, but the horizontal position changes sign. Symmetry allows us to calculate values for an entire revolution knowing only the first quadrant. If $\alpha$ is between $0°$ and $90°$, the corresponding point is $(\cos\alpha,\operatorname{sen}\alpha)$. Reflect it on the axes and in the center:

October 19, 2026 Read

Fundamental relationships: the pieces fit together

In the interactive book: complete chapter III, with all demonstrations and exercises. Two measures that combine Mark a point with coordinates $(3/5,4/5)$ in the first quadrant. Its horizontal and vertical displacements form the legs of a right-angled triangle. The distance to the center is 1. This is a point in the cycle because $(3/5)^2+(4/5)^2=9/25+16/25=1$. Demonstration: the fundamental identity The Pythagorean theorem says that, in a right-angled triangle, the square of the hypotenuse is the sum of the squares of the legs. For a point in the cycle, the lengths of the legs are $|\cos\theta|$ and $|\operatorname{sen}\theta|$, and the hypotenuse measures 1. Squaring eliminates the signs. Therefore, in any quadrant:

October 19, 2026 Read

Circular functions: the wheel that makes waves

In the interactive book: complete chapter II, with all demonstrations and exercises. The Ferris wheel cabin Imagine a Ferris wheel with a center at height zero and a radius equal to one. As the cabin rotates, its height changes. Its horizontal position also changes. Let’s give names to these two measurements. A point on the cycle has coordinates $P=(x,y)$. For the turning angle $\theta$, we define $\cos\theta=x$ and $\operatorname{sen}\theta=y$. Cosine is the horizontal position; sine, the signed height. On the left side, $x$ is negative; below the center, $y$ is negative.

October 19, 2026 Read

Arcs and angles: the mathematics of turns

In the interactive book: complete chapter I, with all demonstrations and exercises. Start with a turn Stand facing a wall and turn until you are facing the opposite wall. You turned around. Another half turn and you are in the starting position. An angle measures the opening between two rays of the same origin. We can also follow the rotation that takes one of them to the other. Take a paper disk. Mark the center O and two points A and B on the edge. Segments OA and OB are rays. The curved path from A to B is an arc. There are two paths along the edge between different points: say which one you are using. The straight segment AB is a chord, different from the arc.

October 19, 2026 Read