Derive Law of Cosines using Pythagorean Theorem

Consider the triangle ABC. Denote h the altitude through B and D the point where h intersects the (extended) base AC
Cosine function for triangle ADB.

cos α= x/c  =>  x=c*cos α
 

Pythagorean theorem for triangle ADB
x2+h2=c2*x2+h2=c2
h2=c2−x2*h2=c2−x2

Pythagorean theorem for triangle CDB
(b−x)2+h2=a2*(b−x)2+h2=a2

Substitute h2 = c2 - x2
(b−x)2+(c2−x2)=a2(b−x)2+(c2−x2)=a2
(b2−2bx+x2)+(c2−x2)=a2(b2−2bx+x2)+(c2−x2)=a2
b2−2
bx+c2=a2b2−2bx+c2=a2

Substitute x = ccos α
b2−2b
(ccosα)+c2=a2b2−2b(c*cos α)+c2=a2

Rearrange to get Law of Cosines

a2=b2+c2−2bc*cos α

JM
Answered by Jan M. Maths tutor

2997 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

The curve A (y = x3 – x2 + x -1) is perpendicular to the straight-line B at the point P (5, 2). If A and B intersect at P, what is the equation of B? Also, find any stationary points of the curve A.


How do you find the coordinate of where two lines intersect?


Find the derivative of A^4 + 2A^2 - 3A + 4


How do you differentiate (3x+cos(x))(2+4sin(3x))?


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences