Trigonometric Ratios in Right-Angled Triangles
Explore how to find acute angles in right-angled triangles using trigonometric ratios. Understand the concepts of opposite, adjacent, and hypotenuse sides, along with sine and cosine ratios for various angles. Dive into the world of trigonometry with practical examples and clear explanations.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
3 July, 2025 Finding angles using trigonometric ratios LO: Use trigonometric ratios to find the acute angles of a right-angled triangle. www.mathssupport.org
Right-angled triangles A right-angled triangle contains a right angle. The right-angled triangle has vertices at the points A, B and C. The angles at these vertices are called ?, ?,and ? respectively. A The side AB, is the longest side, is opposite the right angle, is called the hypotenuse. c b a C B As convention the side BC, opposite to the angle ? is labelled a. the side AC, opposite to the angle ? is labelled b. the side AB, opposite to the angle ? is labelled c. www.mathssupport.org
The opposite and adjacent sides The two shorter sides of a right-angled triangle, generally called legs, are named with respect to one of the acute angles. The side opposite the marked angle is called the opposite side. O P P O S I T E HY P OTE NU S E The side between the marked angle and the right angle is called the adjacent side. A D J A C E N T If we mark this angle www.mathssupport.org
The opposite and adjacent sides The two shorter sides of a right-angled triangle, generally called legs, are named with respect to one of the acute angles. If we mark this angle The side between the marked angle and the right angle is called the adjacent side. A D J A C E N T HY P OTE NU S E O P P O S I T E The side opposite the marked angle is called the opposite side. www.mathssupport.org
Trigonometric ratios Look at this two right-angled triangles. ABCand DEFeach have angles measuring 63o, 90o and 27o DEF is larger than ABC. D 63o A 63o 27o 27o B C F E Triangles with the same three angles are called similar triangles, and their corresponding sides are in the same proportions. For ABCand DEF: =?? ?? =?? ?? =?? ?? ?? ?? ?? ?? ?? and and www.mathssupport.org ??
Trigonometric ratios If we consider any right-angled triangle. ?? ?? hypotenuse ?? =?? opposite = sin = D O P P O S I T E H Y A P H O O P P O S I T E YPO T ENUS TE N US E E C B E F And we mark this angle the length of the opposite side the length of the hypotenuse The ratio of is the sine ratio. www.mathssupport.org
Trigonometric ratios If we consider any right-angled triangle. adjacent hypotenuse ?? ?? ??=?? = cos = D H A Y P H O YPO T ENUS TE N US E E C B E F A D J A C E N T A D J A C E N T And we mark this angle The ratio of the length of the adjacent side the length of the hypotenuse is the cosine ratio. www.mathssupport.org
Trigonometric ratios If we consider any right-angled triangle. opposite adjacent ?? ?? ??=?? = tan = D O P P O S I T E A O P P O S I T E C B E A D J A C E N T F A D J A C E N T And we mark this angle The ratio of the length of the opposite side the length of the adjacent sideis the tangent ratio. www.mathssupport.org
The three trigonometric ratios Opposite Hypotenuse S O H Sin = H O P P O S I T E Y P O T Adjacent Hypotenuse C A H E Cos = N U S E Opposite Adjacent T O A Tan = A D J A C E N T Remember: S O H C A H T O A You can use trigonometric ratios to find unknown side lengths and angles in right-angled triangles. www.mathssupport.org
Relation between sine, cosine and tangent Finding the ratio between sine and cosine In triangle ABC ? ? sin = sin cos ? ? ? ? =? ? ? ? ? ? cos = A c = tan = b B C a sin cos tan = www.mathssupport.org
Finding angles In the right-angled triangle ABC, if we wish to find the size of an acute angle, we need to know the length of two sides. ? ? =sin-1 ? A sin = If then ? c Which reads the angle with a sine ? b ? ? ? =cos-1 ? If then cos = B C ? a Which reads the angle with a cosine ? ? ? ? =tan-1 ? If then tan = ? Which reads the angle with a tangent ? ? www.mathssupport.org
Finding angles o 11 cm Find to 2 decimal places. 13 cm h First label the given sides We are given the lengths of the sides opposite and hypotenuse, so we use: opposite hypotenuse sin =11 13 = sin 111 13 sin = = 57.80 (to 2 d.p.) www.mathssupport.org
Finding angles a Find to 2 decimal places. 4 cm 6 cm h First label the given sides We are given the lengths of the sides adjacent and hypotenuse, so we use: adjacent hypotenuse cos =4 6 = cos 14 6 cos = = 48.19 (to 2 d.p.) www.mathssupport.org
Finding angles o 4 cm a Find to 2 decimal places. 5 cm First label the given sides We are given the lengths of the sides opposite and adjacent to the angle, so we use: opposite adjacent 4 5 = tan 14 5 tan = tan = = 38.66 (to 2 d.p.) www.mathssupport.org
Thank you for using resources from A close up of a cage Description automatically generated For more resources visit our website https://www.mathssupport.org If you have a special request, drop us an email info@mathssupport.org Get 20% off in your next purchase from our website, just use this code when checkout: MSUPPORT_20 www.mathssupport.org
