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vmbo kader and gthavovwo

SOHCAHTOA

After this simulation you can explain why sin(α), cos(α) and tan(α) depend only on the angle, and not on how big the triangle is.

Level
Mode
Right triangle with angle 30° and hypotenuse 8,00 The adjacent side runs along the bottom, the opposite side stands upright and the hypotenuse runs from angle α to the top. 30° adjacent 6,93 opposite 4,00 hypotenuse 8,00
sin(α)=oppositehypotenuse

Start: α = 30°, hypotenuse 8 → opposite 4, sin(30°) = 0,5.

Angle α (degrees)
Hypotenuse
What do you see? α = 30°. opposite ÷ hypotenuse = sin(α) = 0,5000.

Explanation

Keep the angle at 30° and make the hypotenuse longer. The whole triangle grows, but the opposite side stays exactly half of the hypotenuse: sin(30°) = 0,5. That is because all triangles with the same angles are similar. Their sides all grow by the same factor, so the ratios stay the same.

Change the angle and the ratios do change. That is why the angle alone tells you how the sides relate, and why one known side is enough to work out the others. For calculations with steps, use the SOHCAHTOA calculator. Dutch classes call the same rule SOS CAS TOA. Numbers here use a decimal comma (0,5), as in Dutch schools.

Which ratio do you use?

Stand at angle α and name the sides. The opposite side faces α, the adjacent side touches α (and is not the hypotenuse), and the hypotenuse faces the right angle. Then look at the two sides in the question, the one you know and the one you want. Opposite and hypotenuse: SOH. Adjacent and hypotenuse: CAH. Opposite and adjacent: TOA. In Dutch schools this is taught as SOS CAS TOA. From the other acute angle, opposite and adjacent swap names.

Examples by level

  • vmbo: a 6 m ramp makes an angle of 10° with the ground. The ramp is the hypotenuse and the height is the opposite side, so SOH: 6 · sin(10°) = 1,04 m.
  • havo: a right triangle has opposite side 3 and adjacent side 4. That calls for TOA: tan(α) = 3 ÷ 4 = 0,75, and α = tan⁻¹(0,75) = 36,9°.
  • vwo: the adjacent side is 12 cm and α = 40°. For the hypotenuse, use CAH: 12 ÷ cos(40°) = 15,66 cm.

Tasks

  1. Make the opposite and adjacent sides equally long.
  2. Make tan(α) greater than 1.
  3. Choose a hypotenuse of 6 and make the opposite side 3.
  4. Keep sin(α) at 0,5, but make the triangle as large as possible.

Common mistakes

Wrong

Thinking sin(α) gets bigger when the triangle gets bigger.

Right

At the same angle the opposite side and the hypotenuse grow together. The ratio stays the same.

Wrong

Looking for the opposite side without checking which angle it belongs to.

Right

Opposite and adjacent swap when you look from the other acute angle.

What you need to know

Where sine, cosine and tangent sit in the 2027 central exam syllabus. This is not a full exam list.

vmbo kader and gt · wiskunde · WI/K/6 (6.3)

Meetkunde (geometry) · source syllabus wiskunde vmbo 2027 (Examenblad)

havo · wiskunde B · domain C (C1)

Meetkundige berekeningen (geometric calculations), part “sinus, cosinus en tangens” · source syllabus wiskunde B havo 2027 (Examenblad)

vwo · wiskunde B · domain E (E1)

Meetkunde met coördinaten (geometry with coordinates), part “sinus, cosinus en tangens in een rechthoekige driehoek” · source syllabus wiskunde B vwo 2027 (Examenblad)

Test yourself

Practice with SOHCAHTOA practice.

Work it out

SOHCAHTOA calculator with steps

Questions

Does sin(α) change when the triangle gets bigger?

No. If the angle stays the same, every side grows by the same factor. The ratio opposite ÷ hypotenuse stays the same.

When is tan(α) greater than 1?

When the opposite side is longer than the adjacent side, so for an angle greater than 45°.