Geometry and Trigonometry

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Trigonometry is a vital aspect of mathematics that deals with the relationships between the sides and angles of triangles. It is particularly useful in solving problems involving right and non-right angled triangles. Pythagoras's theorem, a fundamental principle in trigonometry, states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Angles of elevation and depression are used in trigonometry to describe the angle at which an observer looks above or below the horizontal line, respectively.

Key Formulae:
Pythagoras’s Theorem: $$ a^2 + b^2 = c^2 $$
Sine Rule: $$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $$
Cosine Rule: $$ c^2 = a^2 + b^2 - 2ab\cos{C} $$

Example:
Consider a right-angled triangle ABC, where angle B is a right angle, AB = 5 units and BC = 12 units. To find the length of AC (the hypotenuse), apply Pythagoras’s theorem:
$$ AC^2 = AB^2 + BC^2 \\ AC^2 = 5^2 + 12^2 \\ AC^2 = 25 + 144 \\ AC^2 = 169 \\ AC = \sqrt{169} \\ AC = 13 \text{ units} $$

This video on Pythagoras' Theorem is from Revision Village and is aimed at students taking the IB Maths Standard level course.