Trigonometric Ratios of Sub Multiple Angles

If A is an angle, then \(\frac{A}{2}\), \(\frac{A}{3}\), \(\frac{A}{4}\) etc. are called sub - multiple angles of A.

Summary

If A is an angle, then \(\frac{A}{2}\), \(\frac{A}{3}\), \(\frac{A}{4}\) etc. are called sub - multiple angles of A.

Things to Remember

Some properties of matrix multiplication:

(i) Multiplication of matrices is, in general, not commutative, i.e. AB not equal to BA, in general.

(ii) Multiplication of matrices in associative, i.e. if A, B and C are matrices conformable for multiplication, then (AB) C = A (BC).

(iii) Multiplication of matrices is distributive with respect to addition i.e. if A, B and C are matrices conformable for the requisite addition and multiplication, then A (B + C) = AB + AC and (A + B) C = AC + BC.

(iv) If A is a square matrix and I is a null matrix of the same order, then AI = IA = A.

Sin A   2sin \(\frac{A}{2}\) Cos \(\frac{A}{2}\)
Cos A Cos2  \(\frac{A}{2}\) - Sin2 \(\frac{A}{2}\)
Cos A 2cos2  \(\frac{A}{2}\) - 1
Cos A 1 - 2 sin2 \(\frac{A}{2}\)
1 + Cos A 2cos2 \(\frac{A}{2}\)
1 - Cos A 2 sin2 \(\frac{A}{2}\)

 

MCQs

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Subjective Questions

Q1:

What are seas regarded as?


Type: Short Difficulty: Easy

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Answer: <p>Seas are regarded as the storehouse of minerals and many precious substances. It has a variety of minerals. These minerals are raw materials for industries. Tourism can be developed if there are sea shores. So, those countries in the world which are developed and progressed are connected to seas and oceans.</p>

Q2:

Mention any four problems faced by Nepal.


Type: Short Difficulty: Easy

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Answer: <p>As Nepal is a landlocked country, it faces various problems. Some of them are as follows:</p>
<ul>
<li>Transportation Problem<br />Nepal has to invest a large amount of capital for roads and airways.</li>
<li>Trade Problem<br />Nepal requires a large amount of capital to export its manufactured and produced goods.</li>
<li>Cultural Problem<br />There is less chance of cultural exchange. There is less opportunity to understand each others' culture and lifestyle.</li>
<li>Economic Problem<br />Most of the landlocked countries are poor. Nepal is one of the poorest&nbsp;countries in the world.</li>
</ul>
<p>&nbsp;</p>
<p>&nbsp;</p>

Q3:

How does price of goods get hiked?


Type: Very_short Difficulty: Easy

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Answer: <p>A landlocked country has to suffer from frequent price hike because of the weak international trade. The country has to depend on the higher amount of imported goods.</p>

Q4:

Why are European landlocked countries relatively developed than the Asian landlocked countries?


Type: Short Difficulty: Easy

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Answer: <p>Many European countries have easy access to waterways. Sea is the sources of minerals. These minerals are raw material for industries. Most of the countries near seas are rich. Tourism can be developed if there are sea shores. Sea beaches are good tourist destinations. Similarly, most of the Asian Countries, depend on expensive means of transport. The cost of airways and roadways are expensive than waterways. So, the European landlocked countries are relatively developed than the Asian landlocked countries.</p>

Q5:

List the problems faced by landlocked countries.


Type: Short Difficulty: Easy

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Answer: <p>A country which is surrounded by land is called a landlocked country. Some of the problems faced by the landlocked countries are as follows:</p>
<ul>
<li>Land transportation costs many times higher than sea transport.</li>
<li>There may be less chance of cultural exchange.</li>
<li>A landlocked country is probably isolated from the rest of the world.</li>
<li>Due to weak international trade, a landlocked country has to suffer from the frequent price hike.</li>
</ul>

Q6:

How can the problem faced by Nepal be solved?


Type: Short Difficulty: Easy

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Answer: <p>Nepal is a landlocked country. It has to face many difficulties. So, to solve the problem, the Government should give proper attention to the country's situation. The country should use the available resources and use a natural protection for its domestic industries, to overcome this condition.</p>

Q7:

Why do landlocked countries bear high transit costs?


Type: Short Difficulty: Easy

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Answer: <p>The landlocked countries bear high transit costs due to physical distance.</p>

Q8:

Why do landlocked countries bear trade deficit?


Type: Short Difficulty: Easy

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Answer: <p>Landlocked countries have to depend&nbsp;on the higher amount of imported goods. So, they have to bear trade deficit.</p>

Videos

Problems faced by landlocked countries
Problems faced by landlocked countries
Problems of Landlocked countries.
Trigonometric Ratios of Sub Multiple Angles

Trigonometric Ratios of Sub Multiple Angles

If A is an angle, then \(\frac{A}{2}\), \(\frac{A}{3}\), \(\frac{A}{4}\) etc. are called sub - multiple angles of A. In this section we wilol discuss about the trigonometric ratios of angle A in terms of \(\frac{A}{2}\) and \(\frac{A}{3}\) .

1. Trigonometric ratios of angle A in terms of \(\frac{A}{2}\)

(a) SinA = sin(\(\frac{A}{2}\) + \(\frac{A}{2}\)) = sin (2 .\(\frac{A}{2}\)) = 2 sin\(\frac{A}{2}\) cos\(\frac{A}{2}\)

(b) sinA = 2sin\(\frac{A}{2}\) cos\(\frac{A}{2}\) =\(\frac{2 sin \frac{A}{2} cos \frac{A}{2}}{cos^2 \frac{A}{2} + sin^2 \frac{A}{2}}\) =\(\frac{2tan \frac{A}{2}}{1 + tan^2 \frac{A}{2}}\)

(c) sinA = \(\frac{2 tan \frac{A}{2}}{1 + tan^2 \frac{A}{2}}\) =\(\frac{\frac{2}{cot \frac{A}{2}}}{1 + \frac{1}{cot^2 \frac{A}{2}}}\) = \(\frac{2 cot \frac{A}{2}}{1 + cot^2 \frac{A}{2}}\)

(d) cosA = cos(2. \(\frac{A}{2}\)) = cos\(^2\)\(\frac{A}{2}\) - sin\(^2\)\(\frac{A}{2}\)

(e) cosA = cos2\(\frac{A}{2}\) - sin\(^2\)\(\frac{A}{2}\) = 1 - sin2\(\frac{A}{2}\) - sin\(^2\)\(\frac{A}{2}\)  = 1 - 2sin\(^2\)\(\frac{A}{2}\)

(f) cosA = cos2\(\frac{A}{2}\) - sin\(^2\)\(\frac{A}{2}\) = cos2\(\frac{A}{2}\) - 1 + cos2\(\frac{A}{2}\) = 2 cos2\(\frac{A}{2}\) - 1

(g) cosA = cos2\(\frac{A}{2}\) - sin\(^2\)\(\frac{A}{2}\) =\(\frac{cos^2 \frac{A}{2} - sin^2 \frac{A}{2}}{cos^2 \frac{A}{2} + sin^2 \frac{A}{2}}\) = \(\frac{1 - tan^2 \frac{A}{2}}{1 + tan^2 \frac{A}{2}}\) (By dividing numerator and denominator by cos2 \(\frac{A}{2}\))

(h) cosA = \(\frac{1 - tan^2 \frac{A}{2}}{1 + tan^2 \frac{A}{2}}\) = \(\frac{1 - \frac{1}{cot^2 \frac{A}{2}}}{{1 + \frac{1}{cot^2 \frac{A}{2}}}}\) = \(\frac{cot^2 \frac{A}{2} - 1}{cot^2 \frac{A}{2} + 1}\)

(i) tanA = tan(2. \(\frac{A}{2}\)) = \(\frac{2 tan\frac{A}{2}}{1 - tan^2 \frac{A}{2}}\)

(j) tanA = \(\frac{2 tan\frac{A}{2}}{1 - tan^2 \frac{A}{2}}\) = \(\frac{\frac{2}{cot \frac{A}{2}}}{1 - \frac{1}{cot^2 \frac{A}{2}}}\) = \(\frac{2 cot \frac{A} {2}}{cot^2 \frac{A}{2} - 1}\)

(k) cotA = cot(2 . \(\frac{A}{2}\)) = \(\frac{cot^2 \frac{A}{2} - 1}{2 cot \frac{A}{2}}\)

(l) cotA = \(\frac{cot^2 \frac{A}{2} - 1}{2cot \frac{A}{2}}\) = \(\frac{\frac{1}{tan^2 \frac{A}{2}} - 1}{tan \frac{A}{2}}\) = \(\frac{1 - tan^2 A}{2 tan \frac{A}{2}}\)

2. Some useful results

(a) 1 + cosA = 1 + cos2 \(\frac{A}{2}\) - sin2 \(\frac{A}{2}\) = 1 - sin2 \(\frac{A}{2}\) + cos2 \(\frac{A}{2}\) = cos2 \(\frac{A}{2}\) + cos2 \(\frac{A}{2}\) = 2 cos2 \(\frac{A}{2}\)

(b) 1 - cosA = 1 - (cos2 \(\frac{A}{2}\) - sin2 \(\frac{A}{2}\)) = 1 - cos2 \(\frac{A}{2}\) + sin2\(\frac{A}{2}\) = sin2 \(\frac{A}{2}\) + sin2 \(\frac{A}{2}\) = 2 sin2 \(\frac{A}{2}\)

(c) 1 + sinA = cos2 \(\frac{A}{2}\) + sin2 \(\frac{A}{2}\) + 2 sin \(\frac{A}{2}\) cos \(\frac{A}{2}\) = (cos \(\frac{A}{2}\) + sin \(\frac{A}{2}\))2

(d) 1 - sinA = cos2\(\frac{A}{2}\) + sin2\(\frac{A}{2}\) - 2 sin v cos\(\frac{A}{2}\)= (cos \(\frac{A}{2}\) - sin \(\frac{A}{2}\))2

3. Trigonometric ratios of A in terms of \(\frac{A}{3}\)

(a) cosA = cos (3 .\(\frac{A}{3}\)) = 3 cos \(\frac{A}{3}\) - 4cos \(^3\) \(\frac{A}{3}\)

(b) sinA = sin (3. \(\frac{A}{3}\)) = 3 sin \(\frac{A}{3}\)- 4 sin\(^3\) \(\frac{A}{3}\)

(c) tanA = tan (3. \(\frac{A}{3}\)) = \(\frac{3 tan \frac{A}{3} - tan^3 \frac{A}{3}}{1 - 3 tan^2 \frac{A}{3}}\)

S. N. Multiple Angle formulae Sub - Multiple Angle formulae
1 sin2A = 2 sinA. cosA sinA = 2sin \(\frac{A}{2}\). cos \(\frac{A}{2}\)
2 sin2A = \(\frac{2 tanA}{1 + tan^2 A}\) sinA =\(\frac{2 tan A \frac{A}{2}}{1 + tan^2 \frac{A}{2}}\)
3 sin2A =\(\frac{2cot A}{1 + cot^2 A}\) sinA =\(\frac{2 cot \frac{A}{2}}{1 + cot^2 \frac{A}{2}}\)
4 cos2A = cos2A - sin2A cosA = cos2 \(\frac{A}{2}\) - sin2 \(\frac{A}{2}\)
5 cos2A = 2cos2A - 1 cosA = 2cos2 \(\frac{A}{2}\) - 1
6 cos2A = 1 - 2 sin2A cosA = 1 - 2sin2\(\frac{A}{2}\)
7 cos2A =\(\frac{1 - tan^2 A}{1 + tan^2 A}\) cos A = \(\frac{1 - tan^2 \frac{A}{2}}{1 + tan^2 \frac{A}{2}}\)
8 cos2A =\(\frac{cot^2 A - 1}{cot^2 A + 1}\) cosA =\(\frac{cot^2 \frac{A}{2} - 1}{cot^2 \frac{A}{2} =+ 1}\)
9 tan2a =\(\frac{2 tanA}{1 - tan^2A}\) tanA =\(\frac{2tan \frac{A}{2}}{1 - tan^2 \frac{A}{2}}\)
10 tan2A =\(\frac{2cot A}{cot^2 - 1}\) tanA =\(\frac{2 cot \frac{A}{2}}{cot^2 \frac{}A{2} - 1}\)
11 cot2A =\(\frac{cot^2 A - 1}{2 cot A}\) cotA =\(\frac{cot^2 \frac{A}{2} - 1}{2 co \frac{A}{2}}\)
12 cot2A =\(\frac{1 - tan^2 A}{2 tanA}\) cotA =\(\frac{1 - tan^2 \frac{A}{2}}{2 tan \frac{A}{2}}\)
13 13sin3A = 3sinA - 4sin\(^3\)A sinA = 3sin \(\frac{A}{3}\) - 4sin \(\frac{A}{3}\)
14 cos3A = 4cos\(^3\)A - 3cosA cosA = 4cos \(^3\) \(\frac{A}{3}\) - 3cos \(\frac{A}{3}\)
15 tan3A =\(\frac{3tanA - tan^3 A}{1 - tan^2 A}\) tanA =\(\frac{3 tan \frac{A}{3} - tan^3\frac{A}{3}}{1 - 3tan^2 \frac{A}{3}}\)
16 1 + cos2a = 2cos2A 1 + cosA = 2cos2 \(\frac{A}{2}\)
17 1 - cos2A = 2sin2A 1 - cosA = 2sin2 \(\frac{A}{2}\)
18 1 + sin2A = ( cosA + sinA )2 1 + sinA = (cos \(\frac{A}{2}\)\(\frac{A}{2}\))2
19 1 - sin2A = (cosA - sinA)2 1 -sinA = (cos \(\frac{A}{2}\)\(\frac{A}{2}\))2

 

 

Lesson

Trigonometry

Subject

Optional Mathematics

Grade

Grade 10

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