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# Introduction to Trigonometry CBSE NCERT Notes Class 10 Maths Chapter 8 PDF

## Trigonometry

**Trigonometry** is the study of relationships between the sides and angles of a triangle.

**Read More : Real Life Application of Trigonometry**

## Degree and Radian Conversion

To convert an angle in degree into radian, we need to multiply π/180 to the degree.

To convert an angle in radian into degree, we need to multiply 180/π to the degree.

### Class 10 Maths Chapter 8 Examples 1:

Convert 30 degree into radian.

30 degree = 30 x π/180 = π/6 radian.

### Class 10 Maths Chapter 8 Examples 2:

Convert 5π/12 radian into degree.

5π/12 radian = 5π/12 x 180/π = 75 degree.

## Arc Length of Sector of Circle

If an arc makes an angle θ (in radian) at the center of a circle whose radius is r, then the length of an arc of the arc is s = rθ. Hence, we can say that the angle θ = s/r radian.

## Pythagoras Theorem

In a right angled triangle, the sum of the square of its perpendicular and the square of its base is equal to the square of the hypotenuse. This is known as **Pythagorean **or** Pythagoras theorem**.

Thus, in a right triangle, according to the Pythagorean theorem p^{2} + b^{2} = h^{2}.

## Trigonometric Ratios

In trigonometry, the trigonometric ratios of an acute angle in a right triangle express the relationship between the angle and the length of its sides.

The values of the trigonometric ratios of an angle do not vary with the lengths of the sides of the triangle, if the angle remains the same.

1. Sine of θ is denoted as **sinθ** and it is the ratio of perpendicular (p) and hypotenuse (h). It means,

**sinθ = p/h = AB/AC.**

2. Cosine of θ is denoted as **cosθ** and it is the ratio of base (b) and hypotenuse (h). It means,

**cosθ = b/h = BC/AC.**

3. Tangent of θ is denoted as **tanθ** and it is the ratio of perpendicular (p) and base (b). It means,

**tanθ = p/b = AB/BC.**

4. Cosecant of θ is denoted as **cosecθ **or** cscθ** and it is the ratio of hypotenuse (h) and perpendicular (p). It means,

**cosecθ or cscθ = h/p = AC/AB.**

5. Secant of θ is denoted as **secθ** and it is the ratio of hypotenuse (h) and base (b). It means,

**secθ = h/b = AC/BC.**

6. Cotangent of θ is denoted as **cotθ** and it is the ratio of base (b) and hypotenuse (h). It means,

**cotθ = b/p = BC/AB.**

## Important Result of Trigonometric Ratios

There are following results about the trigonometric ratios.

7. Since sinθ = p/h, cosθ = b/h and tanθ = p/b, then tanθ = (p/h) / (b/h). Thus,

**tanθ = sinθ / cosθ**.

8. Since sinθ = p/h, cosθ = b/h and cotθ = b/p, then cotθ = (b/h) / (p/h). Thus,

**cotθ = cosθ / sinθ**.

9. Since sinθ = p/h and cosecθ = h/p, then

**sinθ x cosecθ = 1** or **sinθ = 1/ cosecθ** and **cosecθ = 1/ sinθ**.

10. Since cosθ = b/h and secθ = h/b, then

**cosθ x secθ = 1** or **cosθ = 1/ secθ** and **secθ = 1/ cosθ**.

11. Since tanθ = p/b and cotθ = b/p, then

**tanθ x cotθ = 1** or **tanθ = 1/ cotθ** and **cotθ = 1/ tanθ**.

### Class 10 Maths Chapter 8 Examples 3:

Given that tan A = 4/3 , find the other trigonometric ratios of the angle A.

Since tanA = 4/3 = BC/AB, then let AB = 3k and BC = 4k.

Using Pythagoras theorem, we get AC = 5k.

Thus,

sinA = BC/AC = 4/5,

cosA = AB/AC = 3/5,

cosecA = 1/sinA = 5/4,

secA = 1/cosA = 5/3,

cotA = 1/tanA = 3/4

### Class 10 Maths Chapter 8 Examples 4:

If ∠ B and ∠ Q are acute angles such that sin B = sin Q, then prove that ∠ B = ∠ Q.

Let two right triangles ABC and PQR as below, where sin B = sin Q.

Given that sin B = sin Q ⇒ AC/AB = PR/PQ = k. Then, Now

Now, AC/AB = PR/PQ = BC/QR.

Thus, by SSS Similarity, we get Δ ACB ~ Δ PRQ and therefore, ∠ B = ∠ Q.

## Trigonometric Ratios of Some Specific Angles

### Class 10 Maths Chapter 8 Examples 5:

If sin (A – B) = 1/2, cos (A + B) = 1/2, 0° < A + B ≤ 90°, A > B, find A and B.

Since sin (A – B) = 1/2 ⇒ A – B = 30° and cos (A + B) = 1/2 ⇒ A + B = 60°

Solving these two equations, we get A = 45° and B = 15°.

## Trigonometric Ratios of Complementary Angles

The angles whose sum is 90 degree, then both are called complementary angles.

There are following trigonometric ratios for complementary angles.

**sin(90 ^{o} – θ) = +cosθ**

**cos(90 ^{o} – θ) = +sinθ**

**tan(90 ^{o} – θ) = +cotθ**

**cosec(90 ^{o} – θ) = +secθ**

**sec(90 ^{o} – θ) = +cosecθ**

**cot(90 ^{o} – θ) = +tanθ**

### Class 10 Maths Chapter 8 Examples 6:

If sin 3A = cos (A – 26°), where 3A is an acute angle, find the value of A.

sin 3A = cos (90° – 3A) ⇒ cos (90° – 3A) = cos (A – 26°) ⇒ 90° – 3A = A – 26° ⇒ A = 29°.

## Trigonometric Identities

Using Pythagorean theorem p^{2} + b^{2} = h^{2 }and the basic definition of trigonometric ratios, we get three trigonometric identities as below.

**sin ^{2}θ + cos^{2}θ = 1 **

**sec ^{2}θ – tan^{2}θ = 1 **

**cosec ^{2}θ – cot^{2}θ = 1**

### Class 10 Maths Chapter 8 Examples 7:

Prove that secA (1 – sin A)(sec A + tan A) = 1.

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