Introduction to Trigonometry .

Class X

Subject: Mathematics 

The ratios of sides in a right angle triangle is said to be trigonometric ratio.
The ratios of sides are as follows.
1. Perpendicular to hypotenuse
2. Base to hypotenuse
3. Perpendicular to base
4. Base to perpendicular
5. Hypotenuse to base
6. Hypotenuse to perpendicular

These ratios of sides are generally denoted by
1. sine (sin)
2. cosine (cos)
3. tangent (tan)
4. cotangent (cot)
5. secant (sec)
6. cosecant (cosec)

1.The opposite side of considering angle is identified as perpendicular.
2. The adjacent side of considering angle is identified as base.
3. The opposite side of right angle is identified as hypotenuse.

The right triangle ABC in which B is right angle. The considering angle can be either A or C.
When considering angle is C.
1.Perpendicular will be AB.
2. Base will be BC.
3. Hypotenuse will be AC.

When considering angle is A.
1. Perpendicular will be BC.
2. Base will be AB.
3. Hypotenuse will be AC.

There will be no change in the value of hypotenuse in both cases.

When considering angle is A.
The value of trigonometric ratios will be as follows.
1. sinA = BC/AC
2. cosA = AB/AC
3. tanA = BC/AB
4. cotA = AB/BC
5. secA = AC/AB
6. cosecA = AC/BC

When considering angle is C.
The value of trigonometric ratios will be as follows.

1. sinC = AB/AC
2. cosC = BC/AC
3. tanC = AB/BC
4. cotC = BC/AB
5. secA = AC/BC
6. cosecA = AC/AB

Here, the sum of angles A and C will be complementary or right angle.
Therefore,
A = 90⁰ - C
C = 90⁰ - A.
So,
sinA = cosC
cosA = sinC
tanA = cotC
cotA = tanC
secA = cosecC
cosecA = secC

We consider some special trigonometric ratios of 0⁰,30⁰, 45⁰, 60⁰ and 90⁰.

1.For 45⁰, we consider in an  isosceles triangle.
2. For 30⁰ and 60⁰ consider in an equilateral triangle.
3. For 0⁰ and 90⁰ coinsides the perpendicular and base with hypotenuse respectively.

All relations which are derived by Pythagoras theorem is said to be trigonometrical identities which is true for all values of considering angles.

In Right Angle Triangle ABC, in which angle B is right angle.

If considering angle is C which is θ.

AB² + BC² = AC²
When both sides divided by AC².

1. sin²θ + cos²θ = 1

This is a trigonometrical identity and true for all values of θ.

In this way, we find out rest other trigonometrical identities by dividing both sides by BC² and AC² respectively.

Rest others are as follows
4. 1 + tan²θ = sec²θ
5. sec²θ - tan²θ = 1
6. sec²θ - 1 = tan²θ
7. 1 + cot²θ = cosec²θ
8. cosec²θ - cot²θ = 1
9. cosec²θ - 1 = cot²θ

Others trigonometrical identities are derived from reciprocal of ratio of sides and complementary angle of rest angles of triangle.

As per reciprocal,
1. sinθ.cosecθ = 1
2. cosθ.secθ = 1
3. tanθ.cotθ = 1

As per complementary angle.

1. sinθ = cos(90⁰ - θ)
2. cosθ = sin(90 - θ)
3. tanθ = cot(90⁰ - θ)
4. cotθ = tan(90⁰ - θ)
5. secθ = cosec(90⁰ - θ)
6. cosecθ = sec(90⁰ - θ)

These are trigonometrical identities and true for each values of considering angle in right angle triangle.

2. Acids, Bases, and Salts Class X

Learning Points:

1. Acids, Bases, and Salts.

2. Chemical properties acids and base.

3. How to acids and Bases react with metal.

4. Acids and Bases react with each other.

5. Reaction metallic Oxides with acids.

6. Reaction of non metallic Oxides with base.

7. What do all acids and all bases have in common.

8. What happens to an Acids or a Base in water solution.

9. How strong are acid or base solution.

10. Importance of pH in everyday life.

11. Family of salts

12. pH of salts

13. Chemicals from common salts

Questions - Answers

NCERT Textbook

Page No. 2.

Question:

You have been provided with three test tubes. One of them contains distilled water and the other two contain an acidic solution and a basic solution, respectively. If you are given only red litmus paper, how will you identify the contents of each test tube?

Answer

We dip the litmus paper one by one in each test tubes.

If there is no change in the colour of litmus paper we say that solutions can be either distilled water or acidic.

Now, we dip the litmus paper in the third test tube in which colour is changed and we clear that this is basic solution. Now, the colour of litmus paper becomes blue.

Now, we dip the litmus paper in other two test tubes. If colour is changed we say it is acidic otherwise distilled water.

In this process we identify the solutions of test tubes.

Page No.6.

Questions

1. Why should curd or sour substances not be kept in brass and copper vessel?

Answer:

The curd or sour substances are acidic. Copper and brass are matal and alloy of metal. When curd or sour substances reacts with metal displace hydrogen gas and form salt together with harmful products.

Acid + metal    -> hydrogen + salt .

2HSO₄ + 2Cu -> 2CuSO₄ + H₂

Therefore, curd or sour subtances shoud not kept in brass and brass or copper vessel.

Q.No.2. Which gas is usually librated when an acid reacts with a metal? Illustrate with an example. How will you test for presence of this gas?

Answer :

Hydrogen gas is librated during the chemical reaction between acids and salts.

Example is given below

Zn + 2H₂SO₄ -> Zn(SO₄) + 2H₂.

We can test the evolved hydrogen gas by its burning with a pop sound when candle is brought near the gas.

Q.No.3. Metal compound A reacts with dilute hydrochloric acid to produce effervescence. The gas evolved extinguishes a burning candle. Write a balanced chemical equation for the reaction if one of the compound formed is calciumchloride.

Answer:

The balanced chemical equation is given below.

CaCO₃ + 2HCl -> CaCl₂ + CO₂ + H₂O