Cosine Law

Cosine law is a formula used to find the third of SAS triangle. A triangle with two sides given and an included angle. Solving problems that requires law of cosine as solution is easy to recognize.


This derivation of cosine law is graphical method. Given an oblique triangle and put that triangle inside a rectangular plane.

Given a triangle ABC above where side AB is on positive x-axis. By dropping a perpendicular line trough C we create triangle ACD which is a right triangle.

Let y be the distance from point C to D. Let x be the distance from point A to D.

Once we hear a right triangle in trigonometry, that is always associated with SOH-CAH-TOA formula.

In \triangle ACD we have the following relations.



Rearranging both equations we have,



Observe that (x,y) is the coordinate of C. Thus, (x,y)=(bcosA,bsinA).

Also the coordinate of B is (c,0).

Applying the distance formula between B and C we have,




Rearranging the expression we have,


By factoring the common factor b^2


But cos^2A+sin^2A=1


Given a triangle below,

Cosie Law




Consider if the triangle is a right triangle where C=90°,






This reduces to the famous Pythagorean Theorem.

If the SAS triangle ( triangle with two adjacent side and included angle) is given and ask for the third side. Kill the problem using cosine law.

There are problems that you need to know the basic facts about geometry to unveil the hidden given.


“Radii of the same circle are congruent”

“Each side of regular polygon is equal”

Worked Problem 1:

Find the length of each side of regular octagon if the length of the longest diagonal is 10 cm.



The longest diagonal of a polygon is drawn from one vertex to another vertex passing the center of the given polygon. Why? ( there is a circle lurking around the octagon and the longest diagonal is the diameter of the circle )



Using cosine law,



s=5\sqrt{2-\sqrt{2}}=3.83 cm

Worked Problem 2:

Find the length of the rectangle if the length of the diagonal is 20 cm and the diagonal intersects at an angle of 60°.



The side opposite of the larger angle is also the longer side. Hence, the side opposite 120° is the length of the rectangle.

Using cosine law,




L=10\sqrt{3} cm

Sample Problem 3: MMC 2007 Elimination Round

Two points M and N are on the opposite sides of a river. C is point on the same side of the river such that \angle{MCN}=60°. If MC=52 m and CN=72 m. How far M is from N?



Using cosine law,



\overline{MN}=4\sqrt{259} m

Sample Problem 4: Adapted from Canadian Open Mathematics Challenge

\Delta ABC is a right triangle with right angle at C. D is on \overline{AB} such that segment \overline{CD} divides \Delta ABC into two triangles of equal perimeter. If \overline{BC}=15 and \overline{AC}=8, how long is \overline{CD}?



By Pythagorean Theorem,





Let AD=a then, BD=17-a.

Perimeter_{\Delta ADC}=Perimeter_{\Delta DCB}




Also, in \Delta ABC



In \Delta ACD,








Blogger and a Math enthusiast. Has no interest in Mathematics until MMC came. Aside from doing math, he also loves to travel and watch movies.

Latest posts by Dan (see all)

You may also like...

80 Responses

  1. Beula says:

    Can I just say what a relief to locate an individual who basically knows what theyre talking about on the net. You undoubtedly know tips on how to bring an concern to light and make it important. A lot more people should read this and have an understanding of this side of the story. I cant believe youre not additional well-liked because you undoubtedly have the gift.

  2. Hey I know this is off topic but I was wondering if you knew of any widgets I could add to my blog that automatically tweet my newest twitter updates. I’ve been looking for a plug-in like this for quite some time and was hoping maybe you would have some experience with something like this. Please let me know if you run into anything. I truly enjoy reading your blog and I look forward to your new updates.

  3. GUQ14U tout est dans la formation video !. I was actually moved enough to create a thought I

  4. Thanks again for the blog.Really thank you! Really Cool.

  5. Fantastic blog article.Much thanks again. Keep writing.

  6. A round of applause for your post.Really looking forward to read more. Want more.

  7. with us. аА аЂ leаА аЂаse stay us up to dаА аЂаte like thаАаб‚Т€Т“s.

  8. sofa pillows says:

    I am so grateful for your blog post.Really thank you! Awesome.

  9. Wow that was odd. I just wrote an really long comment but after I clicked submit my comment didn at show up. Grrrr well I am not writing all that over again. Anyhow, just wanted to say superb blog!

  10. Thank you ever so for you blog article.Thanks Again.

  11. Remodel says:

    Awesome blog post.Really looking forward to read more. Fantastic.

  12. this this web site conations in fact pleasant funny data

  13. This piece of writing about Search engine marketing is genuinely fastidious one, and the back links are actually very helpful to market your site, its also referred to as SEO.

  14. JD Purity Most wanted Joomla design now available for Drupal

  15. Very good post.Really thank you! Keep writing.

  16. Really informative article.Really looking forward to read more. Awesome.

  17. candles says:

    Looking forward to reading more. Great blog article. Really Great.

  18. UoN says:

    There is noticeably a bundle to know about this. I assume you produced particular nice points in attributes also.

  19. Historie says:

    Thank you ever so for you blog post.Really looking forward to read more. Keep writing.

  20. This is very interesting, You are a very skilled blogger.

  21. Cheap jacket says:

    Thanks, I ave recently been searching for facts about this subject for ages and yours is the best I ave found so far.

  22. Im grateful for the article.Thanks Again. Will read on…

  23. It is actually a great and helpful piece of information. I am happy that you just shared this useful info with us. Please stay us informed like this. Thank you for sharing.

  24. Wow, fantastic blog layout! How long have you been blogging for? you made blogging look easy. The overall look of your site is magnificent, let alone the content!

  25. I think this is a real great blog post.Thanks Again. Cool.

  26. Very good post! We will be linking to this great content on our site. Keep up the great writing.

  27. Thanks for any other fantastic post. Where else may just anybody get that type of info in such a perfect method of writing? I have a presentation next week, and I am at the look for such information.

  28. Angel says:

    Souls in the Waves Excellent Morning, I just stopped in to visit your site and assumed I would say I loved myself.

  29. Your style is really unique in comparison to other folks I have read stuff from. Thank you for posting when you ave got the opportunity, Guess I will just book mark this page.

  30. Wow! Thank you! I always needed to write on my website something like that. Can I take a fragment of your post to my site?

  31. careers says:

    I appreciate you sharing this blog article.Really thank you! Really Great.

  32. I value the article post.Thanks Again. Fantastic.

  33. comida says:

    I will bookmark your blog and check once more right here regularly.

  34. Photography says:

    wow, awesome article post.Thanks Again. Fantastic.

  35. Friv says:

    Outstanding post, you have pointed out some great points, I too conceive this s a very great website.

  36. events says:

    Very informative post.Much thanks again. Really Great.

  37. you will have an amazing weblog here! would you wish to make some invite posts on my weblog?

  38. Thanks so much for the post.Really thank you! Really Great.

  39. You might be my role models. Many thanks to the write-up

  40. Maintenance says:

    Looking forward to reading more. Great post.Thanks Again. Great.

  41. This is really interesting, You are a very skilled blogger. I ave joined your rss feed and look forward to seeking more of your excellent post. Also, I ave shared your web site in my social networks!

  42. Wanted to drop a remark and let you know your Feed isnt working today. I tried adding it to my Yahoo reader account but got absolutely nothing.

  43. Wow! This can be one particular of the most useful blogs We ave ever arrive across on this subject. Actually Wonderful. I am also an expert in this topic therefore I can understand your hard work.

  44. ckmndakos cxbpv ylzejdp ulfk gbyxuioxqhjvewe

  45. This can be an awesome website. and i desire to visit this just about every day from the week.

  46. Within the event you all be able to email myself by incorporating suggestions in how you have made your website search this brilliant, I ad personally have fun right here.

  47. Wonderful article! We will be linking to this particularly great content on our site. Keep up the good writing.

  48. Wow! Thank you! I continuously needed to write on my blog something like that. Can I take a part of your post to my site?

  49. Very good post.Really thank you! Really Cool.

  50. It as hard to come by educated people for this topic, but you sound like you know what you are talking about! Thanks

  51. in the early hours in the dawn, because i love to gain knowledge of more and more.

  52. Thanks again for the post.Much thanks again. Really Great.

  53. Way cool! Some extremely valid points! I appreciate you penning this write-up plus the rest of the site is also really good.

  54. This is one awesome post.Really thank you! Want more.

  55. Is not it excellent if you acquire a fantastic post? Is not it fantastic after you locate a superb post? Supporting the weblog.. thankfully Enjoying the posting.. thanks a bunch

  56. Go Here says:

    This is a topic that is near to my heart Take care! Exactly where are your contact details though?

  57. I value the blog post.Thanks Again. Cool.

  58. This very blog is definitely entertaining additionally informative. I have picked a lot of interesting tips out of this source. I ad love to visit it again soon. Thanks!

  59. Im thankful for the blog post.Really thank you! Keep writing.

  60. Echo says:

    information with us. Please stay us up to date like this.

  61. teen jobs says:

    Major thankies for the blog.Thanks Again.

  62. Leon says:

    newest information. Also visit my web-site free weight loss programs online, Jeffery,

  63. Im grateful for the article post. Fantastic.

  64. I loved your blog post.Really thank you!

  65. Maddie says:

    Spot on with this write-up, I actually suppose this website needs far more consideration. I all in all probability be once more to read way more, thanks for that info.

  66. monster says:

    Im obliged for the article.Really looking forward to read more. Keep writing.

  67. Tammie says:

    You made some decent points there. I checked on the net to learn more about the issue and found most individuals will go along with your views on this site.

  68. Wow, marvelous blog layout! How long have you been blogging for? you make blogging look easy. The overall look of your website is magnificent, as well as the content!

  69. Awesome post.Thanks Again. Cool.

  70. Artist says:

    wow, awesome blog.Thanks Again. Keep writing.

  71. kymco says:

    you are saying and the way in which during which you say it.

  72. This blog was how do I say it? Relevant!! Finally I ave found something which helped me. Thanks a lot!

  73. single says:

    Enjoyed every bit of your blog. Cool.

  74. #lv says:

    Very informative post.Much thanks again. Want more.

  75. Well I truly enjoyed studying it. This article offered by you is very helpful for correct planning.

  76. You made some decent points there. I looked on the internet for the issue and found most individuals will go along with with your blog.

  77. This article will help the internet viewers for creating new blog or even a weblog from start to end.|

  78. Wow, fantastic weblog format! How lengthy have you ever been blogging for? you made running a blog glance easy. The total glance of your web site is wonderful, let alone the content!

  79. Major thanks for the article.Really looking forward to read more. Want more.

Leave a Reply

Your email address will not be published.