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प्रश्न
Given: cos A = `( 5 )/ ( 13 )`
Evaluate:
- `(sin "A "–cot "A") / (2 tan "A")`
- `cot "A" + 1/cos"A"`
उत्तर
Consider the diagram below:
cos A = `( 5 )/( 13 )`
i.e. `"base"/"hypotenuse" = 5/13`
⇒ `"AB"/"AC" = 5/13`
Therefore, if length of AB = 5x, length of AC = 13x
Since
AB2 + BC2 = AC2 ...[Using Pythagoras Theroem]
(5x)2 + BC2 = (13x)2
BC2 = 169x2 – 25x2
BC2 = 144x2
∴ BC = 12x ...(perpendicular)
Now
tan A = `"perpendicular"/"base" = (12x)/(5x) = 12/5`
sin A = `"perpendicular"/"hypotenuse" = (12x)/(13x) = 12/13`
cot A = `"base"/"perpendicular" =(5x)/(12x) = 5/12`
(i) `(sin "A" – cot"A")/ (2tan "A")`
= `( 12 /13 – 5/12)/(2(12/5)`
= `(79)/(156). (5)/(24)`
= `( 395)/(3744 )`
(ii) cot A + `1/ cos "A"`
= `(5)/(12) + (1)/(5/13)`
= `(5)/(12) + (13)/(5)`
= `(181)/(60)`
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