Trigonometric Ratios and Exact Trigonometric Values Notes - Geometry and Measures - Maths GCSE

[Audio] GCSE Maths – Geometry and Measures Trigonometric Ratios and Exact Trigonometric Values Notes WORKSHEET https://bit.ly/pmt-edu-cc https://bit.ly/pmt-cc This work by PMT Education is licensed under CC BY-NC-ND 4.0 https://bit.ly/pmt-cc https://bit.ly/pmt-cc https://bit.ly/pmt-edu.

[Audio] Trigonometric Ratios and Exact Trig Values Labelling a right-angled triangle When working with right-angled triangles, there are three important ratios between the sides and angles. Before we can understand these, you must be able to label the rightangled triangle correctly. 𝜽𝜽 The hypotenuse, denoted by the letter h, is the longest side of the triangle, opposite the rightangle. The opposite side, o, is the side opposite the angle 𝜃𝜃 which we are working with. The adjacent side, a, is the one adjacent to the angle we are working with, that is not the hypotenuse. Trigonometric Ratios The three trigonometric ratios you must know are for sine, cosine and tangent: 𝐬𝐬𝐬𝐬𝐬𝐬 𝜽𝜽 = 𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐 𝒉𝒉𝒉𝒉𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒉𝒉𝒉𝒉𝒐𝒐𝒐𝒐 𝐜𝐜𝐜𝐜𝐬𝐬 𝜽𝜽 = 𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒐𝒐𝒉𝒉𝒐𝒐 𝒉𝒉𝒉𝒉𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒉𝒉𝒉𝒉𝒐𝒐𝒐𝒐 𝐭𝐭𝐭𝐭𝐬𝐬 𝜽𝜽 = 𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐 𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒐𝒐𝒉𝒉𝒐𝒐 You can remember these ratios by the phrase SOH CAH TOA. The first letter indicates the trigonometric ratio and the second letter represents the length which is divided by the length represented by the third letter. For example, SOH helps us to remember that 𝑆𝑆𝑖𝑖𝑖𝑖 𝜃𝜃 = 𝑜𝑜𝑝𝑝𝑝𝑝𝑜𝑜𝑝𝑝𝑖𝑖𝑝𝑝𝑝𝑝 𝑯𝑯𝑦𝑦𝑝𝑝𝑜𝑜𝑝𝑝𝑝𝑝𝑖𝑖𝑦𝑦𝑝𝑝𝑝𝑝 . Using the trigonometric ratios, you can work out an unknown side or unknown angle in a right-angled triangle. https://bit.ly/pmt-cc https://bit.ly/pmt-cc https://bit.ly/pmt-edu.

[Audio] Example: Work out the length of side 𝑥𝑥 in the right-angled triangle ABC, to 3 significant figures. 1. Label the sides of the triangle according to the known angle and decide which trigonometric ratio to use depending on which sides we know. Here, we know the length of the hypotenuse, and are trying to find the opposite side. So, we use the sine ratio: sin 𝜃𝜃 = 𝑜𝑜𝑝𝑝𝑝𝑝𝑜𝑜𝑝𝑝𝑖𝑖𝑝𝑝𝑝𝑝 ℎ𝑦𝑦𝑝𝑝𝑜𝑜𝑝𝑝𝑝𝑝𝑖𝑖𝑦𝑦𝑝𝑝𝑝𝑝 2. Substitute our known values into the trigonometric ratio and solve the equation. sin 45 = 𝑥𝑥 3 𝑥𝑥 = 3 × sin 45 = 𝟐𝟐. 𝟏𝟏𝟐𝟐 𝐜𝐜𝐜𝐜 Example: Find the size of angle ABC in the triangle below, to the nearest degree. 1. Label the sides of the triangle according to the unknown angle and decide which trigonometric ratio to use depending on which sides we know. We know the lengths of the adjacent side and hypotenuse, so we know to use the cosine ratio: cos 𝜃𝜃 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑝𝑝𝑖𝑖𝑝𝑝 ℎ𝑦𝑦𝑝𝑝𝑜𝑜𝑝𝑝𝑝𝑝𝑖𝑖𝑦𝑦𝑝𝑝𝑝𝑝 2. Substitute our known values into the cosine ratio. 𝑎𝑎𝑜𝑜𝑝𝑝 𝜃𝜃 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑝𝑝𝑖𝑖𝑝𝑝 ℎ𝑦𝑦𝑝𝑝𝑜𝑜𝑝𝑝𝑝𝑝𝑖𝑖𝑦𝑦𝑝𝑝𝑝𝑝 = 2 5 3. In this example, we are finding an unknown angle. So, we are required to use the inverse function, for which there is a button on the calculator. 𝑎𝑎𝑜𝑜𝑝𝑝 𝜃𝜃 = 2 5 𝜃𝜃 =𝑎𝑎𝑜𝑜𝑝𝑝−1 � 2 5� = 66.4218. . . ° = 𝟔𝟔𝟔𝟔° (to the nearest degree) https://bit.ly/pmt-cc https://bit.ly/pmt-cc https://bit.ly/pmt-edu.

[Audio] Exact trigonometric values You need to be able to find, without a calculator, the exact values of sin 𝜃𝜃 and cos 𝜃𝜃 for 𝜃𝜃 = 0°, 30°, 45°, 60° and 90°, and the exact values for tan 𝜃𝜃 for 𝜃𝜃 = 0°, 30°, 45° and 60°. The exact values you need to know are summarised in the table below: 𝜽𝜽 = 𝟎𝟎 𝟑𝟑𝟎𝟎° 𝟒𝟒𝟒𝟒° 𝟔𝟔𝟎𝟎° 𝟗𝟗𝟎𝟎° 1 1 2 0 √2 cos 𝜃𝜃 1 √3 2 1 sin 𝜃𝜃 0 1 2 √2 √3 2 1 tan 𝜃𝜃 0 1 1 √3 - √3 You don't need to memorise the three middle columns related to 𝜃𝜃 = 30°, 45°, 60° as you can construct the exact values from the following special triangles. Isosceles right-angled triangle, with the two sides of equal length measuring 1 unit, two remaining angles measuring 45°: We can work out the length of the hypotenuse using Pythagoras' Theorem: ℎ𝑦𝑦𝑝𝑝𝑜𝑜𝑝𝑝𝑝𝑝𝑖𝑖𝑦𝑦𝑝𝑝𝑝𝑝2 = 12 + 12 ℎ𝑦𝑦𝑝𝑝𝑜𝑜𝑝𝑝𝑝𝑝𝑖𝑖𝑦𝑦𝑝𝑝𝑝𝑝2 = 2 ℎ𝑦𝑦𝑝𝑝𝑜𝑜𝑝𝑝𝑝𝑝𝑖𝑖𝑦𝑦𝑝𝑝𝑝𝑝 = √2 The trigonometric exact values when 𝜃𝜃 = 45° are then calculated using trigonometric ratios. For example, to find cos 45, since cos 𝜃𝜃 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 ℎ𝑦𝑦𝑦𝑦𝑦𝑦𝑎𝑎𝑎𝑎𝑎𝑎𝑦𝑦𝑦𝑦𝑎𝑎, we have cos 45 = 1 √2. Equilateral triangle, with side lengths measuring 2 units, angles measuring 60°: We draw a perpendicular bisector down the middle to form a right-angled triangle with angles measuring 30°, 60° and 90°. We can work out the length of the bisector using Pythagoras' Theorem: 𝑏𝑏𝑖𝑖𝑝𝑝𝑝𝑝𝑎𝑎𝑝𝑝𝑜𝑜𝑟𝑟2 = 22 − 12 𝑏𝑏𝑖𝑖𝑝𝑝𝑝𝑝𝑎𝑎𝑝𝑝𝑜𝑜𝑟𝑟2 = 3 𝑏𝑏𝑖𝑖𝑝𝑝𝑝𝑝𝑎𝑎𝑝𝑝𝑜𝑜𝑟𝑟 = √3 https://bit.ly/pmt-cc https://bit.ly/pmt-cc https://bit.ly/pmt-edu.

[Audio] Trigonometric Ratios and Exact Trig Values – Practice Questions 1. Find the length of side 𝑥𝑥 to 1 decimal place. 2. Find the size of angle 𝜃𝜃 to 1 decimal place. 3. Find the exact value of sin 30°. 4. Find the exact value of tan 45°. 5. (Higher only) In the following cuboid, find to 3 significant figures: a) the length of side BH b) the value of angle BHF Worked solutions for the practice questions can be found amongst the worked solutions for the corresponding worksheet file. https://bit.ly/pmt-cc https://bit.ly/pmt-cc https://bit.ly/pmt-edu.