Answer: We can arrange the steps with help of below explanation.
Step-by-step explanation:
Here ABC is a triangle,
Draw a perpendicular from BD to side AC ( construction)
where [tex]D\in AC[/tex]
In the right triangle BCD, from the definition of cosine:
cos C =CD/ BC
CD= a cos C
Subtracting this from the side b, we see that
DA= b-acos C
In the triangle BCD, from the definition of sine:
sin C =BD / a
BD = a sin C
In the triangle ADB, applying the Pythagorean Theorem
[tex]c^2 =BD^2 +DA^2[/tex]
Substituting for BD and DA from (2) and (3)
[tex]c^2 =(asinC)^2 +(b-acosC)^2[/tex]
⇒[tex]c^2 =a^2sin ^2C+b^2-2ab cosC+a^2 cos ^2C[/tex] ( On simplification)
⇒[tex]c^2 =a^2sin ^2C+a^2 cos ^2C+b^2-2abcos C[/tex]
⇒[tex]c^2=a^2(sin^2C+cos^2C) +b^2-2abcosC[/tex]
⇒[tex]c^2 = a^2+b^2-2abcosC[/tex] (because,[tex]sin^2\theta+cos^2\theta =1[/tex] )
