कुछ विशिष्ट कोणों के त्रिकोणमितीय अनुपात - Revision history

Ramamurthy: /* 30° और 60° के त्रिकोणमितीय अनुपात */

2024-09-26T14:43:38Z

30° और 60° के त्रिकोणमितीय अनुपात

← Older revision		Revision as of 20:13, 26 September 2024
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	[[Category:त्रिकोणमिति ~~के कुछ अनुप्रयोग~~]]		[[Category:त्रिकोणमिति का परिचय]]
	[[Category:गणित]]		[[Category:गणित]]
	[[Category:कक्षा-10]]		[[Category:कक्षा-10]]

Ramamurthy: /* 30° और 60° के त्रिकोणमितीय अनुपात */

2024-09-26T14:40:20Z

30° और 60° के त्रिकोणमितीय अनुपात

← Older revision		Revision as of 20:10, 26 September 2024
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			[[Category:त्रिकोणमिति के कुछ अनुप्रयोग]]
	[[Category:गणित]]		[[Category:गणित]]
	[[Category:कक्षा-10]]		[[Category:कक्षा-10]]
	~~[[Category:त्रिकोणमिति के कुछ अनुप्रयोग]]~~

Mani: added content

2024-06-07T07:39:43Z

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← Older revision		Revision as of 13:09, 7 June 2024
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	== 30° और 60° के त्रिकोणमितीय अनुपात ==		== 30° और 60° के त्रिकोणमितीय अनुपात ==
	[[File:Triangle -1.jpg\|alt=Fig. 2 - Triangle\|thumb\|चित्र -2 त्रिभुज]]		[[File:Triangle -1.jpg\|alt=Fig. 2 - Triangle\|thumb\|चित्र -2 त्रिभुज]]
	~~Consider an equilateral~~ <math>\bigtriangleup ABC</math>~~. Each angle in an equilateral triangle is~~ <math>60^\circ</math>, ~~therefore~~,<math>\angle A = \angle B =\angle C =60^\circ</math> .		समबाहु <math>\bigtriangleup ABC</math> पर विचार करें। समबाहु त्रिभुज में प्रत्येक कोण <math>60^\circ</math> होता है, इसलिए, <math>\angle A = \angle B =\angle C =60^\circ</math> .

	~~Draw a perpendicular~~ <math>AD</math> ~~from~~ <math>A</math> ~~to the side~~ <math>BC</math> (~~see Fig.~~ 2).		<math>A</math> से भुजा <math>BC</math> तक एक लंब <math>AD</math> खींचिए (चित्र-2 देखें)।

	~~Now~~ <math>\bigtriangleup ABD=\bigtriangleup ACD</math>		अब <math>\bigtriangleup ABD=\bigtriangleup ACD</math>

	~~Therefore~~, <math>BD=DC</math> ~~and~~ <math>\angle BAD = \angle CAD</math> (~~Corresponding Parts of Congruent Triangles~~)		अतः, <math>BD=DC</math> और <math>\angle BAD = \angle CAD</math> (सर्वांगसम त्रिभुजों के संगत भाग)

	<math>\bigtriangleup ABD</math> ~~is a right angled triangle~~ , ~~right angled at <math>D</math> with~~ <math>\angle BAD = 30^\circ</math> ~~and~~ <math>\angle ABD = 60^\circ</math>		<math>\bigtriangleup ABD</math> एक समकोण त्रिभुज है, जो <math>\angle BAD = 30^\circ</math> और <math>\angle ABD = 60^\circ</math> के साथ <math>D</math> पर समकोण है।

	मान लीजिए <math>AB=2a</math> , अत: <math>BC=AC=AB=2a</math>		मान लीजिए <math>AB=2a</math> , अत: <math>BC=AC=AB=2a</math>

Mani: added content

2024-06-07T07:28:31Z

added content

← Older revision		Revision as of 12:58, 7 June 2024
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	== 45° के त्रिकोणमितीय अनुपात ==		== 45° के त्रिकोणमितीय अनुपात ==
	[[File:Right angle triangle.jpg\|alt=Fig.1 Triangle\|thumb\|चित्र -1 त्रिभुज]]		[[File:Right angle triangle.jpg\|alt=Fig.1 Triangle\|thumb\|चित्र -1 त्रिभुज]]
	In <math>\bigtriangleup ABC</math> ~~right angled at~~ <math>B</math> , If <math>\angle A =45^\circ</math>, <math>\angle C =45^\circ</math>		<math>\bigtriangleup ABC</math> में <math>B</math> समकोण है, यदि <math>\angle A =45^\circ</math>, <math>\angle C =45^\circ</math>

	<math>BC=AB=a</math>		<math>BC=AB=a</math>

	~~Using Pythagoras Theorem~~		पाइथागोरस प्रमेय का उपयोग करते हुए

	<math>AB^2+BC^2=AC^2</math>		<math>AB^2+BC^2=AC^2</math>
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	<math>\bigtriangleup ABD</math> is a right angled triangle , right angled at <math>D</math> with <math>\angle BAD = 30^\circ</math> and <math>\angle ABD = 60^\circ</math>		<math>\bigtriangleup ABD</math> is a right angled triangle , right angled at <math>D</math> with <math>\angle BAD = 30^\circ</math> and <math>\angle ABD = 60^\circ</math>

	~~Let~~ <math>AB=2a</math> , ~~Hence~~ <math>BC=AC=AB=2a</math>		मान लीजिए <math>AB=2a</math> , अत: <math>BC=AC=AB=2a</math>

	<math>BD=\frac{1}{2}BC=\frac{1}{2} \times 2a=a</math>		<math>BD=\frac{1}{2}BC=\frac{1}{2} \times 2a=a</math>
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	<math>cosec \ 30^\circ = \frac{1}{sin \ 30^\circ}=2</math> , <math>sec \ 30^\circ = \frac{1}{cos \ 30^\circ}=\frac{2}{\sqrt{3}}</math> , <math>cot \ 30^\circ = \frac{1}{tan \ 30^\circ}=\sqrt{3}</math>		<math>cosec \ 30^\circ = \frac{1}{sin \ 30^\circ}=2</math> , <math>sec \ 30^\circ = \frac{1}{cos \ 30^\circ}=\frac{2}{\sqrt{3}}</math> , <math>cot \ 30^\circ = \frac{1}{tan \ 30^\circ}=\sqrt{3}</math>

	~~Similarly~~		इसी प्रकार

	<math>sin \ 60^\circ = \ \frac{AD} {AB} =\frac{a\sqrt{3}} {2a}=\frac{\sqrt{3}} {2}</math> , <math>cos \ 60^\circ = \ \frac{BD} {AB} =\frac{a} {2a}=\frac{1} {2}</math> , <math>tan \ 60^\circ = \ \frac{AD} {BD} =\frac{a\sqrt{3}} {a}=\sqrt{3} </math>		<math>sin \ 60^\circ = \ \frac{AD} {AB} =\frac{a\sqrt{3}} {2a}=\frac{\sqrt{3}} {2}</math> , <math>cos \ 60^\circ = \ \frac{BD} {AB} =\frac{a} {2a}=\frac{1} {2}</math> , <math>tan \ 60^\circ = \ \frac{AD} {BD} =\frac{a\sqrt{3}} {a}=\sqrt{3} </math>

Mani: added content

2024-06-07T07:02:51Z

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← Older revision		Revision as of 12:32, 7 June 2024
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	~~In this section~~, ~~we will find the values of the trigonometric ratios for angles of~~ <math>0^\circ ,30^\circ , 45^\circ, 60^\circ , 90^\circ		इस अनुभाग में, हम <math>0^\circ ,30^\circ , 45^\circ, 60^\circ , 90^\circ


	</math>.		</math>के कोणों के लिए त्रिकोणमितीय अनुपातों के मान ज्ञात करेंगे।

	== ~~Trigonometric Ratios of~~ 45° ==		== 45° के त्रिकोणमितीय अनुपात ==
	[[File:Right angle triangle.jpg\|alt=Fig.1 Triangle\|thumb\|~~Fig.~~1 ~~Triangle~~]]		[[File:Right angle triangle.jpg\|alt=Fig.1 Triangle\|thumb\|चित्र -1 त्रिभुज]]
	In <math>\bigtriangleup ABC</math> right angled at <math>B</math> , If <math>\angle A =45^\circ</math>, <math>\angle C =45^\circ</math>		In <math>\bigtriangleup ABC</math> right angled at <math>B</math> , If <math>\angle A =45^\circ</math>, <math>\angle C =45^\circ</math>

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	<math>cosec \ 45^\circ = \frac{1}{sin \ 45^\circ}=\sqrt{2}</math> , <math>sec \ 45^\circ = \frac{1}{cos \ 45^\circ}=\sqrt{2}</math> , <math>cot \ 45^\circ = \frac{1}{tan \ 45^\circ}=1</math>		<math>cosec \ 45^\circ = \frac{1}{sin \ 45^\circ}=\sqrt{2}</math> , <math>sec \ 45^\circ = \frac{1}{cos \ 45^\circ}=\sqrt{2}</math> , <math>cot \ 45^\circ = \frac{1}{tan \ 45^\circ}=1</math>

	== ~~Trigonometric Ratios of~~ 30° ~~and~~ 60° ==		== 30° और 60° के त्रिकोणमितीय अनुपात ==
	[[File:Triangle -1.jpg\|alt=Fig. 2 - Triangle\|thumb\|~~Fig.~~ 2 ~~Triangle~~]]		[[File:Triangle -1.jpg\|alt=Fig. 2 - Triangle\|thumb\|चित्र -2 त्रिभुज]]
	Consider an equilateral <math>\bigtriangleup ABC</math>. Each angle in an equilateral triangle is <math>60^\circ</math>, therefore,<math>\angle A = \angle B =\angle C =60^\circ</math> .		Consider an equilateral <math>\bigtriangleup ABC</math>. Each angle in an equilateral triangle is <math>60^\circ</math>, therefore,<math>\angle A = \angle B =\angle C =60^\circ</math> .

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	{\| class="wikitable"		{\| class="wikitable"
	\|+~~Trigonometric ratios of~~ 0°, 30°, 45°, 60° ~~and~~ 90°		\|+0°, 30°, 45°, 60° और 90° के त्रिकोणमितीय अनुपात
	!<math>\angle A</math>		!<math>\angle A</math>
	!<math>0^\circ</math>		!<math>0^\circ</math>
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	\|<math>1</math>		\|<math>1</math>
	\|<math>\sqrt{3} </math>		\|<math>\sqrt{3} </math>
	\|~~Not Defined~~		\|अपरिभाषित
	\|-		\|-
	\|<math>cosec \ A</math>		\|<math>cosec \ A</math>
	\|~~Not Defined~~		\|अपरिभाषित
	\|<math>2</math>		\|<math>2</math>
	\|<math>\sqrt{2} </math>		\|<math>\sqrt{2} </math>
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	\|<math>\sqrt{2} </math>		\|<math>\sqrt{2} </math>
	\|<math>2</math>		\|<math>2</math>
	\|~~Not Defined~~		\|अपरिभाषित
	\|-		\|-
	\|<math>cot \ A</math>		\|<math>cot \ A</math>
	\|~~Not Defined~~		\|अपरिभाषित
	\|<math>\sqrt{3} </math>		\|<math>\sqrt{3} </math>
	\|<math>1</math>		\|<math>1</math>

Mani at 06:56, 7 June 2024

2024-06-07T06:56:04Z

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 [[Category:गणित]]
 [[Category:कक्षा-10]]
 [[Category:त्रिकोणमिति के कुछ अनुप्रयोग]]

Mani at 12:05, 23 August 2023

2023-08-23T12:05:32Z

← Older revision		Revision as of 17:35, 23 August 2023
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	[[Category:कक्षा-10]]		[[Category:कक्षा-10]]
	[[Category:त्रिकोणमिति के कुछ अनुप्रयोग]]		[[Category:त्रिकोणमिति के कुछ अनुप्रयोग]]
			Trigonometric ratios of some specific Angles

Mani: Updated Category

2023-08-05T06:48:08Z

Updated Category

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	~~[[Category:त्रिकोणमिति]]~~
	[[Category:गणित]]		[[Category:गणित]]
	[[Category:कक्षा-10]]		[[Category:कक्षा-10]]
			[[Category:त्रिकोणमिति के कुछ अनुप्रयोग]]

Mani: Updated Category

2023-08-05T06:43:04Z

Updated Category

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	[[Category:त्रिकोणमिति]]		[[Category:त्रिकोणमिति]]
	[[Category:~~त्रिकोणमिति का परिचय~~]]		[[Category:गणित]]
			[[Category:कक्षा-10]]

Mani: added category

2023-04-21T11:46:57Z

added category

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	[[Category:त्रिकोणमिति]]		[[Category:त्रिकोणमिति]]
			[[Category:त्रिकोणमिति का परिचय]]