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)
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.
Now that you are familiar with the law of cosines you can easily ace all the geometry sections in your boards and entrance exams!
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