Introduction to Trigonometry: Sine, Cosine and Tangent Without Getting Stuck

Trigonometry was born from a practical need: measuring distances and heights that could not be reached directly. How did ancient Greeks calculate the height of a pyramid without climbing it? How did navigators determine their ship's position before GPS? The answer lies in the relationships between angles and sides of triangles.

The name comes from Greek: trigonon (triangle) + metron (measurement). The central idea is that in similar triangles, the ratios between corresponding sides are always equal, regardless of the triangle's size. A right triangle with a 30° angle always has the same proportions between its sides, whether microscopic or gigantic.

Today, trigonometry is everywhere: civil engineering (calculating forces in structures), astronomy (distances between stars), GPS (which uses trigonometric functions), physics (waves, oscillations), and computer graphics (rotating objects in 3D). Understanding the fundamentals opens the door to all these applications.

Basic trigonometry happens in the right triangle — the one with a 90° angle. The three sides have specific names that depend on which reference angle you are using. Imagine you are 'sitting inside' the triangle at the reference angle α.

The hypotenuse is always the side opposite the right angle (90°) — it is the longest side of the triangle. The opposite leg (opposite side) is the side directly across from angle α. The adjacent leg (adjacent side) is the side next to angle α, touching it, but which is not the hypotenuse.

These names change depending on which angle you choose as the reference. If α is at vertex A, the opposite and adjacent legs are determined relative to A. If you switch to angle B of the same triangle, what was the opposite leg for A becomes the adjacent leg for B. That is why you must always mark the reference angle before identifying the sides.

The sine of an angle α is defined as the ratio of the length of the opposite leg to the length of the hypotenuse: sin(α) = opposite leg / hypotenuse. This ratio is always a number between 0 and 1 for acute angles, since the opposite leg can never be larger than the hypotenuse.

Concrete example: if a right triangle has an opposite leg of 3 cm and a hypotenuse of 5 cm, then sin(α) = 3/5 = 0.6. This means angle α has a sine of 0.6. Using a calculator, we discover that α ≈ 36.87°.

Sine can also be used in reverse: if you know the angle and one side, you can find the other. If α = 30° and the hypotenuse is 10 cm, then opposite leg = 10 × sin(30°) = 10 × 0.5 = 5 cm.

The cosine of α is the ratio of the adjacent leg to the hypotenuse: cos(α) = adjacent leg / hypotenuse. While sine focuses on the leg facing the angle, cosine focuses on the leg beside it.

The tangent of α is the ratio of the opposite leg to the adjacent leg: tan(α) = opposite leg / adjacent leg. Note that tan(α) = sin(α) / cos(α). Unlike sine and cosine, tangent can take any real value — including values greater than 1 and very large values near 90°.

A useful mnemonic: SOH-CAH-TOA. Sine = Opposite / Hypotenuse (SOH), Cosine = Adjacent / Hypotenuse (CAH), Tangent = Opposite / Adjacent (TOA). This acronym is widely used and helps remember the three definitions without confusion.

Three angles appear constantly in trigonometry and are worth memorising: 30°, 45° and 60°. For these angles, the values of sine, cosine and tangent can be calculated exactly, without approximations.

For 30°: sin(30°) = 1/2 = 0.5; cos(30°) = √3/2 ≈ 0.866; tan(30°) = 1/√3 ≈ 0.577. For 45°: sin(45°) = cos(45°) = √2/2 ≈ 0.707; tan(45°) = 1. For 60°: sin(60°) = √3/2 ≈ 0.866; cos(60°) = 1/2 = 0.5; tan(60°) = √3 ≈ 1.732.

A way to derive these values without memorising the table is to remember two special triangles. An equilateral triangle cut in half generates a triangle with angles 30°-60°-90°, with sides 1, 2 and √3. An isosceles right triangle has angles 45°-45°-90°, with sides 1, 1 and √2. From these two triangles, all special values emerge.

The protocol for any basic trigonometry problem is: (1) Draw the triangle and mark the reference angle. (2) Identify which sides are known and which must be found. (3) Choose the trigonometric ratio that relates the two sides involved. (4) Set up the equation and solve.

Example 1: a 6-metre ladder leans against a wall making a 60° angle with the ground. At what height does it touch the wall? The reference angle is 60°, the ladder is the hypotenuse (6 m), and we want the opposite leg (the height). Use sine: sin(60°) = height/6. So height = 6 × sin(60°) = 6 × √3/2 = 3√3 ≈ 5.20 metres.

Example 2: on flat ground, you are 50 metres from a building and measure the elevation angle to the top as 35°. What is the building's height? The reference angle is 35°, the horizontal distance is the adjacent leg (50 m) and the height is the opposite leg. Use tangent: tan(35°) = height/50. So height = 50 × tan(35°) ≈ 50 × 0.700 = 35 metres.