Thirty and sixty, cut from an equilateral
An equilateral triangle has three equal sides. Cut it in half.
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- An equilateral triangle. Every side is 2.
- Drop the height. Mark the right angle.
- Mark 30° at the top and 60° at the base.
- Pythagoras gives the height: √3.
- Read six ratios: three at 30°, three at 60°.
One cut, two angles
The height runs from the top corner to the base. It meets the base at a right angle. It cuts the triangle into two halves that match. So the base splits into two parts of 1. The top 60° angle splits into two 30° angles.
In one half, the hypotenuse is 2. The legs, beside the right angle, are 1 and the height. Pythagoras says leg² + leg² = hypotenuse². So height² + 1² = 2².
Then height² = 4 − 1 = 3. The height is √3. The half-triangle has sides 1, √3 and 2.
At the 30° corner, the opposite side is 1. The adjacent side is √3. The hypotenuse is 2.
Rule: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3. sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3.
| Angle | sin | cos | tan |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 60° | √3/2 | 1/2 | √3 |
At the 60° corner, opposite and adjacent swap. The opposite side is √3. The adjacent side is 1. So sin 60° = cos 30° and cos 60° = sin 30°. Tan 60° is the flip of tan 30°.