# 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.

**Derivation:**

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

*be the distance from point A to D.*

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

In 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

But

Given a triangle below,

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

cos90°=0

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.

Facts:

*“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.

Solution:

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,

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°.

Solution:

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,

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 °. If and . How far M is from N?

Solution:

Using cosine law,

**Sample Problem 4: Adapted from Canadian Open Mathematics Challenge**

is a right triangle with right angle at . is on such that segment divides into two triangles of equal perimeter. If and , how long is ?

Solution:

By Pythagorean Theorem,

Let then, .

Also, in

In ,

### Dan

#### Latest posts by Dan (see all)

- 2016 MMC Schedule - November 4, 2015
- 2014 MTAP reviewer for Grade 3 - September 30, 2015
- 2015 MTAP reviewer for 4th year solution part 1 - August 22, 2015

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.

I’m also writing to let you know of the cool experience my daughter had checking yuor web blog. She learned several issues, including what it’s like to possess an amazing giving mindset to get most people without difficulty fully understand chosen tortuous issues. You really exceeded people’s expected results. Thank you for churning out such great, safe, educational and also unique tips on the topic.

I am now not positive the place you’re getting your information, however good topic. I needs to spend some time learning more or understanding more. Thanks for great information I used to be in search of this information for my mission.