Trigonometry Basics Explained — SOHCAHTOA & Pythagoras’ Theorem (JAMB/WAEC Maths)
Trigonometry is the branch of mathematics that studies the relationships between the angles and sides of triangles, particularly right-angled triangles. This topic — covering the meaning of trigonometry, its basic ratios (SOHCAHTOA), and Pythagoras’ theorem — is foundational in JAMB and WAEC Mathematics, since nearly every later trigonometry topic builds directly on these basics. This lesson explains each concept with worked examples.
Quick takeaways
- Trigonometry studies the relationships between the angles and sides of triangles, especially right-angled triangles.
- The three basic trigonometric ratios are sine, cosine, and tangent, commonly remembered using the mnemonic SOHCAHTOA.
- SOHCAHTOA stands for: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
- Pythagoras’ theorem, a² + b² = c², relates the three sides of a right-angled triangle, where c is the hypotenuse.
- Correctly labeling the opposite, adjacent, and hypotenuse sides relative to the angle in question is the essential first step before applying any trigonometric ratio.
What is trigonometry?
Trigonometry is the branch of mathematics concerned with the relationships between the angles and side lengths of triangles, most commonly applied to right-angled triangles at the JAMB/WAEC level. It allows you to calculate an unknown angle or side length in a triangle when certain other measurements are already known, and it has wide practical applications, from construction and navigation to physics.
How do you correctly label the sides of a right-angled triangle?
Before applying any trigonometric ratio, you must correctly identify three sides relative to a specific angle (usually marked θ) within the right-angled triangle:
- Hypotenuse — the longest side, always opposite the right angle, and this labeling never changes regardless of which angle you’re working with
- Opposite — the side directly across from the angle θ you’re focusing on
- Adjacent — the side next to angle θ (that is not the hypotenuse)
Since “opposite” and “adjacent” depend on which angle you’re focusing on, correctly identifying the reference angle first is essential — getting this step wrong will lead to using the wrong ratio entirely.
What is SOHCAHTOA, and what does it stand for?
SOHCAHTOA is a mnemonic used to remember the three basic trigonometric ratios:
- Sine = Opposite / Hypotenuse (SOH)
- Cosine = Adjacent / Hypotenuse (CAH)
- Tangent = Opposite / Adjacent (TOA)
These three ratios allow you to find a missing side or angle in a right-angled triangle, depending on which two pieces of information you already have. For example, if you know the angle and the hypotenuse, and need to find the opposite side, you would use the sine ratio.
What is Pythagoras’ theorem?
Pythagoras’ theorem describes the relationship between the three sides of any right-angled triangle:
a² + b² = c²
Where c is the hypotenuse (the longest side), and a and b are the other two sides. This theorem allows you to find the length of any one side of a right-angled triangle, provided you know the lengths of the other two — and unlike SOHCAHTOA, it doesn’t require knowing any angle at all, only the side lengths.
How do you decide whether to use SOHCAHTOA or Pythagoras’ theorem?
The choice generally depends on what information you’re given and what you need to find:
- Use Pythagoras’ theorem when you know (or need to find) side lengths only, with no angles involved
- Use SOHCAHTOA when the problem involves an angle, either as given information or as what you need to calculate
Some problems require combining both — for example, using Pythagoras’ theorem to find a missing side first, then using SOHCAHTOA to find an angle.
Common mistakes students make with trigonometry basics
- Mislabeling the opposite and adjacent sides by not clearly identifying the reference angle first
- Using the wrong trigonometric ratio for the given information, rather than checking which sides/angle are actually known
- Forgetting that the hypotenuse is always the longest side and always opposite the right angle, regardless of which other angle is being used as reference
- Attempting to use SOHCAHTOA when only side lengths are known and no angle is involved, when Pythagoras’ theorem is the appropriate tool
Frequently asked questions
Q: What does SOHCAHTOA stand for? SOHCAHTOA stands for Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, and Tangent = Opposite/Adjacent — the three basic trigonometric ratios.
Q: What is Pythagoras’ theorem used for? Pythagoras’ theorem, a² + b² = c², is used to find a missing side length in a right-angled triangle when the other two side lengths are known, without needing any angle information.
Q: How do you know which side is the “opposite” side in a triangle? The opposite side is the one directly across from the specific angle you’re focusing on — this changes depending on which angle in the triangle you’re using as your reference point.
Q: When should you use SOHCAHTOA instead of Pythagoras’ theorem? Use SOHCAHTOA when the problem involves an angle (either given or to be found), and use Pythagoras’ theorem when the problem involves only side lengths with no angles.
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