Question

Day 7

Angles of a ∆ABC form an increasing ap then find the value of sinB.​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

↝ Angles of a triangle forms an increasing AP series.

↝ Since, as triangle have 3 angles.

As, We know that, 3 numbers in AP series is taken as a - d, a, a + d.

So, Let assume that,

$\purple{\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:\begin{cases} &\sf{\angle A = a - d} \\ \\ &\sf{\angle B = a} \\ \\ &\sf{\angle C = a + d} \end{cases}\end{gathered}\end{gathered}}$

We know, Sum of interior angles of a triangle is supplementary.

Thus,

$\rm \implies\:\angle A + \angle B + \angle C = 180\degree$

$\rm \implies\:a - d + a + a + d = 180\degree$

$\rm \implies\:3a= 180\degree$

$\purple{\bf \implies\:a= 60\degree}$

$\purple{\bf \implies\:\angle B= 60\degree}$

So,

$\green{\rm :\longmapsto\:sin\angle B \: }$

$\green{\rm :\longmapsto\:\boxed{ \tt{ \: sin60\degree = \frac{ \sqrt{3} }{2} \: }}}$

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Additional Information :

↝ nᵗʰ term of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

aₙ is the nᵗʰ term.

a is the first term of the sequence.

n is the no. of terms.

d is the common difference.

↝ Sum of n  terms of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{n}{2} \bigg(2 \:a\:+\:(n\:-\:1)\:d \bigg)}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

Sₙ is the sum of n terms of AP.

a is the first term of the sequence.

n is the no. of terms.

d is the common difference.