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त्रिकोणमिति का परिचय

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Question
CBSEHHIMAH10010205
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ΔABC में, जिसका कोण B समकोण है, AB = 24 cm और BC = 7 cm है। निम्नलिखित का मान ज्ञात कीजिए:
(i) sin A cos A,
(ii) sin C, cos C.

Solution

हमे प्राप्त हैं, AB = 24 cm
BC = 7cm, पाइथागोरस प्रमेय के प्रयोग से,
AC2 = AB2 + BC2
= (24 cm)2+ (7 cm)2
= 576 cm2 + 49 cm2
= 625 cm2
So, AC = 25 cm

अब, 
   (i) sin space straight A space equals space BC over AC space equals space 7 over 25 comma space space space cos space straight A space equals space AB over AC space equals 24 over 25
   (ii) sin space straight C space equals space AB over AC space equals space 24 over 25 comma space space space space cos space straight C space equals space BC over AC equals space 7 over 25

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Question
CBSEHHIMAH10010206
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सलंग्न आकृति में tan p - cot R का मान ज्ञात कीजिए।

Solution

मान लीजिए:
PQ = 12K
तथा 
PR = 13K
पाइथागोरस प्रमेय के प्रयोग से:
PR2 = PQ2 + QR2
⇒(13K)2 = (12K)2 + QR2
⇒169K2 = 144K2 + QR2
⇒QR2 = 169K2 - 144K2
⇒ QR2 = 25K2
∴ QR = 5K

अब,  
tan space straight P space equals space QR over PQ equals fraction numerator 5 straight K over denominator 12 straight K end fraction equals 5 over 12 cot space straight R space equals space QR over PQ equals fraction numerator 5 straight K over denominator 12 straight K end fraction equals 5 over 12
अत:
tanP minus cotR space equals space 5 over 12 minus 5 over 12 equals 0.

Question
CBSEHHIMAH10010207
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यदि sin space straight A space equals space 3 over 4 तो cosA तथा tanA का मान परिकलित कीजिए।

Solution

increment माना कि ABC एक समकोण त्रिभुज है जिसमें angle straight B space equals space 90 degree
हम जानते हैं: 
space sin space straight A space equals space 3 over 4 space equals space BC over AC
 Let BC = 3K,  AC = 4K
यहाँ k एक घनात्मक संख्या हैं। 

पाइथागोरस प्रमेय के प्रयोग से:
                 space space AC squared space equals space AB squared plus BC squared
rightwards double arrow        space space space space space space left parenthesis 4 straight K right parenthesis squared space equals space AB squared plus left parenthesis 3 straight K right parenthesis squared
rightwards double arrow              space 16 straight K squared space equals space AB squared plus 9 straight K squared
rightwards double arrow              space AB squared space equals space 16 straight K squared minus 9 straight K squared
rightwards double arrow               AB squared space equals space 7 straight K squared
rightwards double arrow               AB equals square root of 7 straight K
अब,  cosA space equals space AB over AC equals fraction numerator square root of 7 straight K over denominator 4 straight K end fraction space equals fraction numerator square root of 7 over denominator 4 end fraction
और    tanA space equals space BC over AB equals fraction numerator 3 straight K over denominator square root of 7 straight K end fraction equals fraction numerator 3 over denominator square root of 7 end fraction.

 

Question
CBSEHHIMAH10010208
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यदि 15cot A = 8 हो तो sinA और secA का मान ज्ञात कीजिए।

Solution

increment ABC माना कि एक समकोण त्रिभुज है जिसमें angle straight B space equals space 90 degree
हमें दिया गया हैं:
15 cot A = 8
rightwards double arrow              space space cotA space equals space 8 over 15 equals AB over BC

माना 
AB = 8K, BC = 15K
पाइथागोरस प्रमेय के प्रयोग से:
AC2 = AB2 + BC2 = (8K)2 + (15K)2
= 64K2 + 225K2
= 289K2
अत;
AC = 17K
अब,
AC = 17K
sinA equals BC over AC equals fraction numerator 15 straight K over denominator 17 straight K end fraction equals 15 over 17 secA space equals space AC over AB equals fraction numerator 17 straight K over denominator 8 straight K end fraction equals 17 over 8.

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