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

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Trigonometry+ 02


Problem A

Prove the following trigonometric identities using the Euler formula
ei·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) =
= ei·(x+y) = ei·x·ei·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)
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)