Using-the-cosine-rule-c-2-a-2-b-2-2abcosC-prove-the-triangle-inequality-if-a-b-and-c-are-sides-of-a-triangle-ABC-then-a-b-c-and-explain-when-equality-holds-Further-prove-that-sin-sin-s

Question Number 112059 by Aina Samuel Temidayo last updated on 05/Sep/20

Using the cosine

rule (c^{2} = a^{2} + b^{2} - 2 abcosC), prove the

triangle inequality : if a, b and c are

sides of a triangle ABC, then a + b ⩾ c

and explain when equality holds .

Further prove that \sin α + \sin β ⩾

\sin (α + β) for 0 ° ⩽ α, β ⩽ 180 °

Answered by 1549442205PVT last updated on 06/Sep/20

i) From the cosine theorem we have

c^{2} = b^{2} + a^{2} - 2 abcosC

\Rightarrow a^{2} + b^{2} + 2 ab = c^{2} + 2 abcosC + 2 ab

\Rightarrow {(a + b)}^{2} = c^{2} + 2 ab (1 + cosC) (1)

Since C is an angle of the triangle, 0 < C < 180 °

\Rightarrow cosC > - 1 \Rightarrow 1 + cosC > 0 . Hence,

c^{2} + 2 ab (1 + cosC) > c^{2} (2)

From (1) (2) we infer

{(a + b)}^{2} > c^{2} \Rightarrow a + b > c

ii) We have \sin (α + β) = \sin α \cos β + \cos α \sin β

Since 0 < α, β ⩽ 180 °, \sin α, \sin β > 0 . Also,

\cos α, \cos β ⩽ 1 . Hence

\sin α \cos β ⩽ \sin α (3)

\sin β \cos α ⩽ \sin β (4)

From (3) (4) we get \sin (α + β) =

\sin α \cos β + \cos α \sin β ⩽ \sin α + \sin β (q . e . d)

iii) Prove \sin α + \sin β ⩾ 2 \sin \frac{α + β}{2}

By the convert formula for the sum

of two sine we have

\sin α + \sin β = 2 \sin \frac{α + β}{2} \cos \frac{α - β}{2} (5)

Since 0 < α < 180, 0 < \frac{α + β}{2} < 90 °

\Rightarrow \sin \frac{α + β}{2} > 0 . On the other hands,

\cos \frac{α - β}{2} ⩽ 1 . Hence, 2 \sin \frac{α + β}{2} \cos \frac{α + β}{2}

⩽ 2 \sin \frac{α + β}{2} (6) . From (5) (6) we get

\sin α + \sin β ⩾ 2 \sin \frac{α + β}{2} (q . e . d)

Commented by Aina Samuel Temidayo last updated on 06/Sep/20

From (1) (2)

why do we infer {(a + b)}^{2} > c^{2} . I {don}^{'} t

understand please .

Commented by Aina Samuel Temidayo last updated on 06/Sep/20

For (ii), please how did \sin α + \sin β ⩾

\sin (α + β) change into \sin α + \sin β

⩾ 2 \sin \frac{α + β}{2}

Commented by 1549442205PVT last updated on 06/Sep/20

I wrote a mistake and corrected

Commented by Aina Samuel Temidayo last updated on 06/Sep/20

I {don}^{'} t get . {What}^{'} s the mistake ?

Commented by Aina Samuel Temidayo last updated on 06/Sep/20

Oh . Thanks . But do I need (iii) ?