Trigonometry+ 02: UNIZOR.COM - Math+ & Problems - Trigonometry

Notes to a video lecture on http://www.unizor.com

Trigonometry+ 02


Problem A

Prove the following trigonometric identities using the Euler formula
e^i·φ = cos(φ)+i·sin(φ)
[See lecture on UNIZOR.COM - Math 4 Teens - Trigonometry - Complex Numbers and Trigonometry - Euler's Formula]

sin(x+y) =
= sin(x)·cos(y) + cos(x)·sin(y)

cos(x+y) =
= cos(x)·cos(y) − sin(x)·sin(y)

Proof A
cos(x+y) + i·sin(x+y) =
= e^i·(x+y) = e^i·x·e^i·y =
= [cos(x) + i·sin(x)] ·
· [cos(y) + i·sin(y)]=
=cos(x)·cos(y)+i²sin(x)·sin(y)+
+i·sin(x)·cos(x)+i·cos(x)·sin(y)=
=cos(x)·cos(y)−sin(x)·sin(y)+
+i·[sin(x)·cos(y)+cos(x)·sin(y)]


Problem B

Prove the following trigonometric identities

tan(x+y) =

tan(x)+tan(y) 1−tan(x)·tan(y)

cot(x+y) =

cot(x)·cot(y)−1 cot(x)+cot(y)

Proof B

tan(x+y) =

sin(x+y) cos(x+y)

=	sin(x)·cos(y) + cos(x)·sin(y) cos(x)·cos(y) − sin(x)·sin(y)

Divide numerator and denominator by cos(x)·cos(y)

=	sin(x)/cos(x) + sin(y)/cos(y) 1−sin(x)·sin(y)/(cos(x)·cos(y))

Cotangent formula is proved analogously.


Problem C

Prove the following trigonometric identities

sin(x) =

2·tan(½x) 1+tan²(½x)

cos(x) =

1−tan²(½x) 1+tan²(½x)

Proof C
sin(x) = sin(½x + ½x)
= 2sin(½x)·cos(½x)
= 2[sin(½x)/cos(½x)]·cos²(½x)
= 2·tan(½x) · cos²(½x)
At the same time
1/[1+tan²(½x)]
= 1/[1+sin²(½x)/cos²(½x)]
= cos²(½x)/[cos²(½x)+sin²(½x)]
= cos²(½x)