Are you stumped by geometry problems involving triangles? Do you want to learn a neat trick to improve your geometry skills? Read on about the law of cosines to do well in your engineering entrance exam.
Basics of trigonometry
Let us revise the basics of trigonometry. Given a right-angled triangle, we can find 6 standard ratios based on the sides of the triangle, which describe the angles.
Consider the triangle ABC as shown above. We see that <C is a right angle. This makes side c the hypotenuse.
Thus, we can find 6 trigonometry ratios that describe <a
sin <a = a/c = opp/hyp
cos <a= b/c
tan <a = a/b
sec <a= c/a
cosec <a =c/b
cot <a = b/a
Similarly, we can find all the trigonometric ratios that describe <b.
Applying trigonometric ratios to non-right angled triangles
We know that we can find these ratios for all angles using log tables and trigonometric formulae. Using log tables is essential to apply the law of cosines. We can use these values to find the values of sides of other triangles as well.
What is the law of cosines?
The law of cosines deals with any given triangle ABC as shown above, and states that:
c2 =a2 + b2 -2abcosC
A more recognisable form is:
cos C= a2 + b2 -c2 / 2ab
Similarly, we can use the law of cosines to find cos A and cos B as well:
cos B= a2 + c2 -b2 / 2ac
cos A= b2 + c2 -a2 / 2bc
Thus, you can now apply the law of cosines in the following situations:
When all sides are known to find the cosines of the angles and by extension, the angles themselves
When two sides and the angle between them is known, used to find the third side, and by extension all the angles in the triangle (Refer to situation 1)
EXAMPLE PROBLEM:
Let there be a triangle ABC with the following values:
We see that a=6, b=10, c=7
Thus, by substituting with the law of cosines formula we get:
cos C = 62 + 102 -72 / (2 * 6 * 10)
= 136-49 /12
= 87 /120
Therefore, <C = cos-1 (87/120) = 43.53 degrees
We can now similarly find all the angles of ABC. Note that we can apply the law of cosines to any triangle with any angle values as long as the rules that make a shape a triangle still hold good.
Further prep
Now that you are familiar with the law of cosines you can easily ace all the geometry sections in your boards and entrance exams!
But what if you have doubts about certain topics and questions? After all, textbooks cannot explain every scenario. To help yourself get the most out of your books, consider taking up coaching. But wait, isn’t coaching expensive and time-consuming? Well, it needn’t be.
While traditional coaching IS expensive and exhausting for you to take up after a long day at school, online coaching isn’t. It is comparatively more affordable and allows you to walk at your own pace, at your own time. To get some expert advice, question papers, mock exams, and overall assistance join an online coaching platform. A popular one is Edureify.
Edureify provides a topic-wise set of questions that are tailored to the exam you choose to appear for. The platform also has a dedicated set of teachers who specialise in each subject, so you can request to be tutored by them to improve your performance.
Edureify solving sets are designed in levels of increasing difficulty designed to test your preparation. In addition to this, be sure to participate in the game-quizzes and AIR challenges that are full of higher-order-thinking questions, mirroring exams like JEE. Edureify even allows you to test yourself based on various parameters: Concept Analysis, and Speed Analysis. Both are important to succeed in your board exams and most entrances.
The recommendation is to get your Concept Analysis to a level you’re happy with, and then Speed Analysis- do not forget that many tests have negative marking, so strong concepts triumph over the speed of solving.
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