#### Pythagoras and Trigonometry

**PYTHAGORAS THEOREM**

- A right-angle triangle is a triangle with one of its angles as 90°.

- The Pythagoras theorem states the square of the hypotenuse of a right-angle triangle is equal to the sum of the square of its sides.
- The side of the triangle that is opposite the right angle is the
*hypotenuse*. - The side of the triangle that is opposite the arc is the
*opposite*. - The last side is called the
*adjacent*. - The hypotenuse is the longest side of any right angle triangle.
- Mathematically: hypotenuse² = opposite² + adjacent²; AB² = BC² + AC²

**PYTHAGOREAN TRIPLES**

- Pythagorean triples are a set of three integers, a, b, c, that describe the sides of a right angle triangle.
- The integers satisfy the Pythagoras theorem; a² + b² = c²; where c is the hypotenuse.
- Examples of Pythagorean triples include:
- 3, 4, 5
- 15, 8, 17

- If a set of Pythagorean triples is multiplied by a factor, the result is also a set of Pythagorean triples. For example:
- (3, 4, 5) × 2 = (6, 8, 10) ⇒ is a Pythagorean triple
- (5, 12, 13) × 3 = (15, 36, 39) ⇒ is a Pythagorean triple

**TRIGONOMETRY**

- Trigonometry basically studies the relationship between side lengths and angles of triangles.
- Sine (sin), cosine (cos), tangent (tan) are functions used in trigonometry.
- The ratios of a right angle triangle are:
- SOH:
**s**in θ =^{opposite}/_{hypotenuse} - CAH:
**c**os θ =^{adjacent}/_{hypotenuse} - TOA:
**t**an θ =^{opposite}/_{adjacent}

- SOH:
- If you cannot use the Pythagoras theorem to solve for the unknown side, as long as you are given an angle, you can use any of the ratios above to solve the problem.

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