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Right triangle trigonometry

Level 4 · Ages 13–15

Quick recall· Let's wake up what you already know before we start.

Recall: in a right triangle, where is the hypotenuse?

The whole lesson

Read it again any time, including while you work through the exercises.

How do you measure the height of a cliff without climbing it? Surveyors have done it since antiquity: measure a distance on the ground, an angle, and trigonometry does the rest. Its favourite playground: the right triangle.

First, learn to name the sides relative to a chosen acute angle (call it angle A). The hypotenuse you know: opposite the right angle, the longest. The side opposite A is the one facing A, without touching it. The side adjacent to A touches A without being the hypotenuse. Careful: if you switch angles, opposite and adjacent swap roles!

Then, the three magic ratios. The sine of A is opposite divided by hypotenuse. The cosine of A is adjacent divided by hypotenuse. The tangent of A is opposite divided by adjacent. To remember them, whisper SOH CAH TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

These ratios depend only on the angle, not on the triangle's size: a small and a large right triangle with the same angle have exactly the same ratios. That is why your calculator knows sin, cos and tan of any angle!

Two uses: if you know an angle and one side, you compute another side. If you know two sides, you recover the angle with the calculator's inverse keys (arcsin, arccos, arctan). Make sure your calculator is in degree mode, and round as the problem asks.

The worked example, step by step

In a right triangle, the hypotenuse is 10 cm and one acute angle is 30°. Find the length of the side opposite this angle.

  1. opposé et hypoténuse : sinus

    I identify the sides involved: the wanted side is OPPOSITE the angle, and I know the HYPOTENUSE. The ratio linking them is the sine (SOH).

  2. sin(30°) = opposé / 10

    I write the sine definition with my values: sine of 30° equals the opposite side divided by 10.

  3. opposé = 10 × sin(30°) = 10 × 0,5 = 5

    I multiply both sides by 10. The calculator (in degree mode!) gives sin(30°) = 0.5, so the side is 5 cm.

Result: The opposite side is 5 cm.

In a right triangle, the side adjacent to angle A is 4 cm and the hypotenuse is 5 cm. Find angle A, rounded to the nearest degree.

  1. cos(A) = adjacent / hypoténuse = 4/5

    Adjacent and hypotenuse: that is the cosine (CAH). I write cos(A) = 4/5 = 0.8.

  2. A = arccos(0,8)

    I know the cosine and want the angle: I use the inverse key arccos (often written cos⁻¹) on the calculator.

  3. A ≈ 36,87° ≈ 37°

    The calculator shows about 36.87. Rounded to the degree: 37°. Consistent: a cosine close to 1 gives a fairly small angle.

Result: Angle A measures about 37°.