Math · Geometry

Sine, Cosine and Tangent Explained Simply

In a right triangle, sine, cosine and tangent describe how the sides relate to an angle α. With them you can calculate sides and angles without measuring them directly.

sin(α) = Opposite ÷ Hypotenuse
cos(α) = Adjacent ÷ Hypotenuse
tan(α) = Opposite ÷ Adjacent Classic mnemonic: SOH-CAH-TOA
α Opposite Adjacent Hypotenuse

đź’¬ Mia asks Grandpa Theo

Mia

How do I remember which side is opposite and which is adjacent?

Grandpa Theo

Look at the angle α you're focusing on: the green side directly across from it is the opposite. The blue side right next to it, that isn't the hypotenuse, is the adjacent. The red hypotenuse is always the longest side, across from the right angle.

Mia

What's this actually used for in real life?

Grandpa Theo

For example, to calculate heights you can't measure directly - like a tree's height, if you only know the angle to its top and your distance from it!

Mia

Show me an example!

Grandpa Theo

A tree casts a 20m shadow when the sun is at a 30° angle. The height is then Adjacent × tan(30°) = 20 × 0.577 ≈ 11.5 meters.

Try it yourself

A ladder (hypotenuse) is 5m long and leans at 60° to the ground. How high does it reach?
Answer: Opposite = 5 × sin(60°) = 5 × 0.866 ≈ 4.33m

You'll find sin, cos and tan values on a calculator (SIN/COS/TAN keys) - you only need to memorize a few standard angles like 30°, 45° and 60°.

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