Fundamental concepts before going through the topic (विषय मार्फत जानु अघि आधारभूत अवधारणाहरू)
Concept 1(अवधारणा १)
- Similar(~) triangles will also follow the same ratio; For Example: In the Figure below figure 1 is in the ratio 3:4:5 and we can compare figure 2 with figure 1 as 6 is the multiple of 3 and 10 is the multiple of 5 and both are multiple of 2. Therefore figure (ii) should also follow the same rule as (समान (~) त्रिभुजहरूले पनि समान अनुपात पछ्याउनेछन्; उदाहरणका लागि: तलको चित्रमा चित्र १ को अनुपात ३:४:५ मा छ र हामीले चित्र १ सँग चित्र २ तुलना गर्न सक्छौँ किनकि ६ को ३ को गुणन हो र १० भनेको ५ को गुणन हो र दुबै २ को गुणन हो। त्यसैले चित्र (ii) ले पनि सोही नियमको पालना गर्नुपर्छ)3*2:4* 2:5*2 = 6:8:10
- Directly the value of remaining portion will be 8 and same continues in other figures. (सीधै बाँकी भागको मूल्य 8 हुनेछ र अन्य अंकहरूमा उही जारी छ)
Concept 2 (अवधारणा २):
- In any right angled triangle longest leg should be hypotenuse.(कुनै पनि समकोण त्रिकोणमा सबैभन्दा लामो खुट्टा कर्ण हुनुपर्छ।)
- Opposite of greater angle in any triangle should have the greater side and smaller will have small side. (कुनै पनि त्रिभुजमा ठूलो कोणको विपरित पक्ष ठूलो र सानोको सानो पक्ष हुनुपर्छ।)
- Oftenly the problems will be asked when the angle of elevation or depression will be 30°, 45° and 60°. (प्रायः समस्याहरू सोधिनेछन् जब उचाइ वा अवसादको कोण 30°, 45° र 60° हुनेछ।)
- In any right angled triangle if one side is 45° then remaining will automatically be 45°(कुनै पनि समकोण त्रिभुजमा यदि एउटा भुजा ४५° छ भने बाँकी स्वतः ४५° हुनेछ)
(i.e. 90°+45°+x=180°; x=45°)
- In any right angled triangle if one side is 60° then remaining will automatically be 30°. (कुनै पनि समकोण त्रिभुजमा यदि एउटा भुजा ६०° छ भने बाँकी स्वतः ३०° हुनेछ।)
Prerequisite
- How to find Square Root? (स्क्वायर रूट कसरी पत्ता लगाउने?)
- Sum of interior angle of triangle is 180°. (त्रिभुजको भित्री कोणको योगफल 180° हो)
- Table given below is very lengthy process and need to remember alot of information. (तल दिइएको तालिका धेरै लामो प्रक्रिया हो र धेरै कुरा सम्झनु आवश्यक छ।)
- Remembering table given below is strictly prohibited.
0° | 30° | 45° | 60° | 90° | Remarks | |
0 | 1 | 2 | 3 | 4 | (write from 0 to 4) | |
√(0/4) | √(1/4) | √(2/4) | √(3/4) | √(4/4) | divide by 4 and make root of each | |
Sin | 0 | 1/2 | 1/√2 | √3/2 | 1 | simplify |
Cos | 1 | √3/2 | 1/√2 | 1/2 | 0 | rotate the simplified value of sin from 90 to 0 |
Tan | 0 | 1/√3 | 1 | √3 | ∞ | divide sin/cos |
Cosec | ∞ | 2 | √2 | 2/√3 | 1 | 1/ sin |
Sec | 1 | 2/√3 | √2 | 2 | ∞ | 1/cos; or rotate cosec 90 to 0 for sec 0 to 90 respectively |
Cot | ∞ | √3 | 1 | 1/√3 | 1 | 1/ tan |
- This will also follow certain rule as shown in the figure below: (तलको चित्रमा देखाइए अनुसार यसले निश्चित नियमहरू पनि पालन गर्नेछ:)
- You can solve all the problems just by remembering figure below:
Example
- Two ships are sailing in the sea on the two sides of a lighthouse. The angle of elevation of the top of the lighthouse is observed from the ships are 30° and 45° respectively. If the lighthouse is 100 m high, the distance between the two ships is: (एउटा लाइटहाउसको दुई छेउमा समुद्रमा दुईवटा जहाज चलिरहेका छन्। लाइटहाउसको शीर्षको उचाइको कोण जहाजबाट अवलोकन गरिन्छ क्रमशः 30° र 45° हो। यदि लाइटहाउस 100 मिटर अग्लो छ भने, दुई जहाजहरू बीचको दूरी हो:)
a. 173 m
b. 200 m
c. 273 m
d. 300 m
In the figure above (माथिको चित्रमा)
- From triangle ABD given angle is 45°; therefore the height and base will be equal. (त्रिभुज ABD बाट दिइएको कोण 45° छ; त्यसैले उचाइ र आधार बराबर हुनेछ।)
- From another triangle ACB angle is 30°; therefore the height will be a= 100 and base =a√3= 100√ 3. Therefore both the ships are (अर्को त्रिकोणबाट ACB कोण 30° हो; त्यसैले उचाइ a=100 र आधार =a√3= 100√ 3 हुनेछ। त्यसैले दुबै जहाजहरू)
100√3 m+100m apart(टाढा)
i.e. more than 200 and less than 300 should be correct option. (अर्थात् 200 भन्दा बढी र 300 भन्दा कम सही विकल्प हुनुपर्छ)
- C is the correct option.(C सही विकल्प हो।)
2. The angle of elevation of a ladder leaning against a wall is 60° and the foot of the ladder is 4.6 m away from the wall. The length of the ladder is:
a. 2.3 m
b. 4.6 m
c. 7.8 m
d. 9.2 m
3. An observer 1.6 m tall is 20 √3 away from a tower. The angle of elevation from his eye to the top of the tower is 30°. The height of the tower is:
a. 21.6 m
b. 23.2 m
c. 24.72 m
d. None
4. From a point P on a level ground, the angle of elevation of the top tower is 30°. If the tower is 100 m high, the distance of point P from the foot of the tower is:
a. 149 m
b. 156 m
c. 173 m
d. 200 m
5. The angle of elevation of the sun, when the length of the shadow of a tree √3 times the height of the tree, is:
a. 30°
b. 45°
c. 60°
d. 90°
Note: Questions will simply be logical and easy to do without calculator.
नोट: प्रश्नहरू केवल तार्किक र क्याल्कुलेटर बिना गर्न सजिलो हुनेछ।
For more clearer view of this topic you can go through this video.(यस विषयको थप स्पष्ट दृश्यको लागि तपाइँ यो भिडियो मार्फत जान सक्नुहुन्छ।)