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Basic Inverse Trigonometric Functions

In Δ ABC ...

Question

In $Δ$ $A B C$ the sides opposite to angles $A, B, C$ are denoted by $a, b, c$ respectively. $r$ is the in-radius of the traingle $A B C$
If $r_{1}, r_{2}, r_{3}$ are in A.P. for the triangle $A B C$ , then $cot (A / 2), cot (B / 2), cot (B / 2)$ are in?

A.P.

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G.P.

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H.P.

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none of these

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Solution

The correct option is C H.P.
since, $r_{1}, r_{2}, r_{3}$ sre in AP then $t a n \frac{A}{2}, t a n \frac{B}{2}, t a n \frac{C}{2}$ are in AP as they are Proportional to each other respectively

and therefore, $c o t \frac{A}{2}, c o t \frac{B}{2}, c o t \frac{C}{2}$ is inversely proportional to $r_{1}, r_{2}, r_{3}$

Hence, they are in $H P$

Therefore, Answer is $C$

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Similar questions

Δ

A B C

the sides opposite to angles

A, B, C

are denoted by

a, b, c

respectively. If

r_{1}

amd

r_{2}

are the ex-radii of the traingle, then

(r_{1} + r_{2}) tan \frac{C}{2}

Δ

A B C

the sides opposite to angles

A, B, C

are denoted by

a, b, c

respectively. Let

r_{1} . r_{2}

and

r_{3}

be the ex-radii of the traingle and

r

be the in-radius.

Find the value of

r_{1} + r_{2} + r_{3} - r

Δ

A B C

the sides opposite to angles

A, B, C

are denoted by

a, b, c

respectively, then

r_{1}^{2} + r_{2}^{2} + r_{3}^{3} + r^{2} = ?

(where

r

=

in-radius, R

=

circumradius,

r_{1}, r_{2}, r_{3}

are ex-radii)

Δ

A B C

the sides opposite to angles

A, B, C

are denoted by

a, b, c

respectively. Let

r_{1}

and

r_{2}

be the ex-radius of the traingle.

In a triangle

A B C, \frac{r_{1} + r_{2}}{1 + cos C}

is equal to?

Δ

A B C

the sides opposite to angles

A, B, C

are denoted by

a, b, c

respectively.

r_{1}, r_{2}, r_{3}

are the ex-radii then if

r_{1} = 2 r_{2} = 3 r_{3}

, the value of

a : b

is equal to?

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In Δ ABC the sides opposite to angles A,B,C are denoted by a,b,c respectively. r is the in-radius of the traingle ABCIf r1,r2,r3 are in A.P. for the triangle ABC, then cot(A/2),cot(B/2),cot(B/2) are in?

The correct option is C H.P.since, r1,r2,r3 sre in AP then tanA2,tanB2,tanC2 are in AP as they are Proportional to each other respectively and therefore, cotA2,cotB2,cotC2 is inversely proportional to r1,r2,r3Hence, they are in HPTherefore, Answer is C

In $Δ$ $A B C$ the sides opposite to angles $A, B, C$ are denoted by $a, b, c$ respectively. $r$ is the in-radius of the traingle $A B C$
If $r_{1}, r_{2}, r_{3}$ are in A.P. for the triangle $A B C$ , then $cot (A / 2), cot (B / 2), cot (B / 2)$ are in?