Trigonometric Ratios Revision Notes

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

# Trigonometric Ratios Trigonometry involves calculating angles and sides in triangles. ## Labelling the Sides The three sides of a right-angled triangle have special names. - The **hypotenuse** (*h*) is the longest side. It is opposite the right angle. - The **opposite side** (*o*) is opposite the angle in question (*x*). - The **adjacent side** (*a*) is next to the angle in question (*x*). **Diagram:** A right-angled triangle with: - Hypotenuse (*h*) labelled on the longest side - Opposite side (*o*) labelled opposite the angle - Adjacent side (*a*) labelled next to the angle ## 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. ### Using an Equilateral Triangle 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. **Diagram:** An equilateral triangle with side lengths of 2 cm, split into two right-angled triangles with: - The base of each triangle labelled 1 cm - The height of each triangle labelled √3 cm - The hypotenuse labelled 2 cm - Angles of 30° and 60° labelled on the triangles 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* **Trigonometric Ratios:** | Angle | Sin | Cos | Tan | |---|---|---|---| | 30° | 1/2 | √3/2 | 1/√3 or √3/3 | | 60° | √3/2 | 1/2 | √3 | ### Using a Square A square with side lengths of 1 cm can be used to calculate accurate values for the trigonometric ratios of 45°. Split the square into two right-angled triangles. **Diagram:** A square with side lengths of 1 cm, split into two right-angled triangles with: - The base of each triangle labelled 1 cm - The height of each triangle labelled 1 cm - The hypotenuse labelled √2 cm - Each angle labelled 45° Calculate the length of the third side of the triangle using **Pythagoras' theorem**: - *a² + b² = c²* - *c² = 1² + 1²* - *c = √2*