Trigonometric Ratios Notes PDF

## Trigonometric Ratios Trigonometry involves calculating angles and sides in triangles. ### Labelling the Sides The three sides of a right-angled triangle have special names: * **Hypotenuse (h):** the longest side. It is opposite the right angle. * **Opposite side (o):** opposite the angle in q...

## Trigonometric Ratios Trigonometry involves calculating angles and sides in triangles. ### Labelling the Sides The three sides of a right-angled triangle have special names: * **Hypotenuse (h):** the longest side. It is opposite the right angle. * **Opposite side (o):** opposite the angle in question (x). * **Adjacent side (a):** next to the angle in question (x). A diagram of a right-angled triangle is provided, showing the sides labelled: * **Hypotenuse (h):** top side of the triangle * **Opposite (o):** side opposite the angle x * **Adjacent (a):** side next to the angle x ### Three Trigonometric Ratios Trigonometry involves three ratios - **sine**, **cosine**, and **tangent**, which are abbreviated to **sin**, **cos**, and **tan**. The three ratios are calculated by calculating the ratio of two sides of a right-angled triangle: * **sin x = opposite / hypotenuse** * **cos x = adjacent / hypotenuse** * **tan x = opposite / adjacent** A useful way to remember these is: **SOH CAH TOA** ## Exact Trigonometric Ratios for 0°, 30°, 45°, 60° and 90° The trigonometric ratios for the angles 30°, 45°, and 60° can be calculated using two special triangles. An equilateral triangle with side lengths of 2 cm can be used to find exact values for the trigonometric ratios of 30° and 60°. The equilateral triangle can be split into two right-angled triangles. A diagram of an equilateral triangle is provided, showing the lengths of the sides and the angles. * **2 cm** - side length of the equilateral triangle * **30°** - angles of the right-angled triangle within the equilateral triangle * **60°** - angles of the right-angled triangle within the equilateral triangle * **√3 cm** - length of the third side of the right-angled triangle (calculated using Pythagoras's theorem) * **1 cm** - base length of the right-angled triangle The length of the third side of the triangle can be calculated using **Pythagoras' theorem**: * a² + b² = c² * a² = c² - b² * a² = 2² - 1² * a² = 3 * a = √3 ### Using the Trigonometric Ratios Use the trigonometric ratios to calculate accurate values for the angles 30° and 60°. | Angle | sin x | cos x | tan x | |---|---|---|---| | 30° | 1/2 | √3 / 2 | 1 / √3 or √3 / 3 | | 60° | √3 / 2 | 1 / 2 | √3 | ## Calculating Trigonometric Ratios for 45° A square with side lengths of 1 cm can be used to calculate accurate values for the trigonometric ratios of 45°. ### Splitting the Square Split the square into two right-angled triangles. A diagram of a square is provided, showing the lengths of the sides and the angles: * **1 cm** - side length of the square * **45°** - angles of the right-angled triangle within the square * **√2 cm** - length of the third side of the right-angled triangle (calculated using Pythagoras's theorem) ### Calculating the Third Side Calculate the length of the third side of the triangle using **Pythagoras' theorem**: * a² + b² = c² * c² = 1² + 1² * c = √2