In any triangle ABC, show thatHence, prove that if sides a,b,c a

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Trigonometric Functions

Short Answer Type

171.

Find $space space space straight capital delta comma space straight r space and space straight R comma$ when $straight a equals 22 comma space straight b equals 28 comma space straight c equals 36$

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172. In any triangle ABC, show that

left parenthesis abc right parenthesis space straight s space equals space straight capital delta squared space cosec straight A over 2 cosec straight B over 2 cosec straight C over 2

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173.

In any triangle, show that: $4 Rcos straight A over 2 cos straight B over 2 cos straight C over 2 equals straight s$

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174. In any triangle ABC, show that

sin space straight A plus sin space straight B plus sin space straight C space equals space fraction numerator straight capital delta over denominator straight r space straight R end fraction

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175. In any triangle ABC, show that

cosA space plus space cosB space plus space cosC space equals space 1 plus straight r over straight R

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Long Answer Type

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176. In any triangle ABC, show that

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Hence, prove that if sides a,b,c are in A.P., $cot straight A over 2 comma space cot straight B over 2 space and space cot straight C over 2$ are in A.P.

space space straight r space equals space straight capital delta over straight s

tan open parentheses straight A over 2 close parentheses space equals space square root of fraction numerator left parenthesis straight s minus straight b right parenthesis space left parenthesis straight s minus straight c right parenthesis over denominator straight s left parenthesis straight s minus straight a right parenthesis end fraction end root

∴

left parenthesis straight s minus straight a right parenthesis space tan space straight A over 2 space equals space left parenthesis straight s minus straight a right parenthesis square root of fraction numerator left parenthesis straight s minus straight b right parenthesis left parenthesis straight s minus straight c right parenthesis over denominator straight s left parenthesis straight s minus straight a right parenthesis end fraction end root space equals space square root of fraction numerator left parenthesis straight s minus straight a right parenthesis left parenthesis straight s minus straight b right parenthesis left parenthesis straight s minus straight c right parenthesis over denominator straight s end fraction end root

equals fraction numerator square root of straight s left parenthesis straight s minus straight a right parenthesis left parenthesis straight s minus straight b right parenthesis left parenthesis straight s minus straight c right parenthesis end root over denominator straight s end fraction space equals space straight capital delta over straight s space equals space straight r

Hence,

left parenthesis straight s minus straight a right parenthesis space tan straight A over 2 space equals space straight r

Similarly,

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and

space space space space space space left parenthesis straight s minus straight c right parenthesis space tan straight C over 2 space equals space straight r

Now,

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are in A.P.

if

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are in A.P.

if

space space fraction numerator 1 over denominator straight r divided by left parenthesis straight s minus straight a right parenthesis end fraction comma space fraction numerator 1 over denominator straight r divided by left parenthesis straight s minus straight b right parenthesis end fraction comma space fraction numerator 1 over denominator straight r divided by left parenthesis straight s minus straight c right parenthesis end fraction

are in A.P.

if

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are in A.P.

if

straight s minus straight a comma space space space straight s minus straight b comma space space straight s minus straight c

are in A.P.

if

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are in A.P.

if a, b, c are in A.P.

which is true (given)

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Short Answer Type

177.

In triangle ABC, if b = 15 cm, $space space cosB space equals space 4 over 5 comma$ find R.

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178.

Convert to radian measure:

$40 degree space 20 apostrophe$

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179.

Convert to radian measure:

$40 degree space 15 apostrophe space 36 apostrophe apostrophe$

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180.

Convert to radian measure:

$22 degree space 30 apostrophe$

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