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Trigonometric Functions

Short Answer Type

601. Deduce the sine formula from cosine formula.

93 Views

Long Answer Type

602. In any triangle ABC, prove that

straight a over sinA space equals space straight b over sinB space equals space fraction numerator straight c over denominator sin space straight C end fraction comma

i.e. sides of the triangle are proportional to the sines of the opposite angles.

103 Views

603. In any triangle ABC, prove that

$cos space straight A space equals space fraction numerator straight b squared plus straight c squared minus straight a squared over denominator 2 bc end fraction comma space cos space straight B space equals space fraction numerator straight c squared plus straight a squared minus straight b squared over denominator 2 ac end fraction space and space cos space straight C space equals space fraction numerator straight a squared plus straight b squared minus straight c squared over denominator 2 ab end fraction$

where a, b and c are as usual the sides of the triangle.

Proof:



Let angle C be acute in figure (i), obtuse in figure (ii) and right angle in figure
(iii) Draw AD $perpendicular$ BC in all the triangles (on produced if necessary)

 $straight c squared space equals space AB squared space equals space BD squared plus AD squared$ ....(i)

In figure (i), $space space AD over AC equals sinC$

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In figure (iii), $AD over AC space equals space 1 space equals space sin 90 degree space equals space sinC$

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∴ In each figure, AD = b sin C ....(ii)

In figure (ii), $BD space equals space BC minus DC space equals space BC minus AC space. DC over AC space equals space straight a minus straight b space cosC$

In figure (i), $space space BD space equals space BC plus DC space equals space BC plus AC. DC over AC equals straight a plus straight b space cos left parenthesis straight pi minus straight C right parenthesis space equals space straight a minus straight b space cosC$

In figure (iii), $BD space equals space BC space equals space straight a minus straight b space cos 90 degree space equals space straight a minus straight b space cosC$

∴ In each figure, $BD space equals space straight a minus straight b space cosC$ ....(iii)

Using (ii) and (iii) in (i), we get

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$rightwards double arrow$ $straight c squared equals straight a squared plus straight b squared cos squared straight C minus 2 ab space cosC plus straight b squared sin squared straight C$

$rightwards double arrow$ $2 ab space cosC space equals space straight a squared plus straight b squared left parenthesis cos squared straight C plus sin squared straight C right parenthesis minus straight c squared$

$rightwards double arrow$ $2 ab space cosC space equals space straight a squared plus straight b squared minus straight c squared$

$rightwards double arrow$ $cosC space equals space fraction numerator straight a squared plus straight b squared minus straight c squared over denominator 2 ab end fraction$

Similarly, $cosA equals fraction numerator straight b squared plus straight c squared minus straight a squared over denominator 2 bc end fraction space space and space space cosB space equals space fraction numerator straight a squared plus straight c squared minus straight b squared over denominator 2 ac end fraction$

Hence proved.

108 Views

604.

In any triangle ABC, prove that:

(i) a = b cos C + c cos B (ii) b = c cos A + a cos C (iii) c = a cos B + b cos A

118 Views

Short Answer Type

605.

Prove that in any triangle ABC.

$space space straight capital delta space equals space 1 half bc space sinA space equals space 1 half ac space sinB space equals space 1 half ab space sinC equals square root of straight s left parenthesis straight s minus straight a right parenthesis left parenthesis straight s minus straight b right parenthesis left parenthesis straight s minus straight c right parenthesis end root$

98 Views

606.

In any triangle ABC, prove that

$space space space straight capital delta space equals space 1 half straight a squared fraction numerator sinB space sinC over denominator sinA end fraction equals 1 half straight b squared fraction numerator sinC space sinA over denominator sin space straight B end fraction space equals space 1 half straight c squared fraction numerator sinA space sinB over denominator sin space straight C end fraction$

101 Views

607.

Prove that:

$straight s left parenthesis straight s minus straight a right parenthesis space tan straight A over 2 equals straight capital delta$

90 Views

Long Answer Type

608.

Prove that for any triangle ABC

$2 straight R space equals space fraction numerator straight a over denominator sin space straight A end fraction equals straight b over sinB equals fraction numerator straight c over denominator sin space straight C end fraction$

where R is the radius of the circumcircle.

89 Views

Short Answer Type

609.

Prove that in any triangle ABC

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