Introduction to Trigonometry

1. Meaning (Definition) of Trigonometry

The word trigonometry is derived from the Greek words ‘tri’ meaning three, ‘gon’ meaning sides and ‘metron’ meaning measure.

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

2. Positive and negative angles

Angle measured in anticlockwise direction is taken as positive angle whereas the angle measured in clockwise direction is taken as negative angle.

3. Trigonometric Ratios

Ratio of the sides of a right triangle with respect to the acute angles is called the trigonometric ratios

of the angle.

Trigonometric ratios of the acute angle A in right triangle ABC are given as follows:

4. Important facts about Trigonometric ratios

• Trigonometric ratios of an acute angle in a right triangle represents the relation between the angle and the sides.
• The ratios defined above can be rewritten as sin A, cos A, tan A, cosec A, sec A and cot A.
• Each trigonometric ratio is a real number and it has not unit.
• All the trigonometric symbols i.e., cosine, sine, tangent, cotangent, secant and cosecant, have no literal meaning.
• (sinθ)ⁿ is generally written as sinⁿ θ, n being a positive integer. Similarly, other trigonometric ratios can also be written.
• The values of the trigonometric ratios of an angle do not vary with the length of the sides of the triangle, if the angles remain the same.

5. Pythagoras theorem

It states that “in a right triangle, square of the hypotenuse is equal to the sum of the squar es of the other two sides”.

Pythagoras theorem can be used to obtain the length of the side of a right angled triangle when the other two sides are already given.

6. Relation between trigonometric ratios

The ratios cosec A, sec A and cot A are the reciprocals of the ratios sin A, cos A and tan A respectively as given:

1. 

2. 

3. 

4. 

5. Values of Trigonometric ratios of some specific angles:

∠A	0°	30°	45°	60°	90°
sin A	0				1
cos A	1				0
tan A	0		1		Not defined
cosec A	Not defined	2			1
sec A	1			2	Not defined
cot A	Not defined		1		0

• The value of sin A or cos A never exceeds 1, whereas the value of sec A or cosec A is always greater than 1 or equal to 1.
• The value of sinθ increases from 0 to 1 when θ increases from 0^o to 90^o.
• The value of cosθ decreases from 1 to 0 when θ increases from 0^o to 90^o.
• If one of the sides and any other parts like either an acute angle or any side of a right triangle are known, the remaining sides and angles of the triangle can be obtained using trigonometric ratios.

8. Trigonometric ratios of complementary angles

Two angles are said to complementary angles if their sum is equal to 90^o. Based on this relation, the trigonometric ratios of complementary angles are given as follows:

1. sin (90^o – A) = cos A
2. cos (90^o – A) = sin A
3. tan (90^o – A) = cot A
4. cot (90^o – A) = tan A
5. sec (90^o – A) = cosec A
6. cosec (90^o – A) = sec A

Note: tan 0^o = 0 = cot 90^o, sec 0^o = 1 = cosec 90^o, sec 90^o, cosec 0^o, tan 90^o and cot 0^o are not defined.

9. Definition of Trigonometric Identity

An equation involving trigonometric ratios of an angle, say θ, is termed as a trigonometric identity if it is satisfied by all values of θ.

10. Basic trigonometric identities

• sin² θ + co s² θ = 1
• 1 + tan² θ = sec² θ; 0 ≤ θ < 90 ^o
• 1 + cot² θ = cosec² θ; 0 ≤ θ < 90 ^o

11. Opposite & Adjacent Sides in a Right Angled Triangle

In the ΔABC right-angled at B, BC is the side opposite to ∠A, AC is the hypotenuse and AB is the side adjacent to ∠A.

12. Trigonometric Ratios

For the right ΔABC, right-angled at ∠B, the trigonometric ratios of the ∠A are as follows:

13. Visualization of Trigonometric Ratios Using a Unit Circle

Draw a circle of the unit radius with the origin as the centre. Consider a line segment OP joining a point P on the circle to the centre which makes an angle θ with the x-axis. Draw a perpendicular from P to the x-axis to cut it at Q.

14. Relation between Trigonometric Ratios

15. Range of Trigonometric Ratios from 0 to 90 degrees

tanθ and secθ are not defined at 90^∘.

cotθ and cosecθ are not defined at 0^∘.

16. Variation of trigonometric ratios from 0 to 90 degrees

17. Standard values of Trigonometric ratios

18. Complementary Trigonometric ratios

In Mathematics, the complementary angles are the set of two angles such that their sum is equal to 90°. For example, 30° and 60° are complementary to each other as their sum is equal to 90°. In this article, let us discuss in detail about the complementary angles and the trigonometric ratios of complementary angles with examples in a detailed way.

If θ is an acute angle, its complementary angle is 90^∘ − θ. The following relations hold true for trigonometric ratios of complementary angles.

Finding Trigonometric Ratios of Complementary Angles

∠A and ∠C form a complementary pair.

⇒ ∠A + ∠C = 90°

The relationship between the acute angle and the lengths of sides of a right-angle triangle is expressed by trigonometric ratios. For the given right angle triangle, the trigonometric ratios of ∠A is given as follows:

The trigonometric ratio of the complement of ∠A. It means that the ∠C can be given as 90° – ∠A

As ∠C = 90°- A (A is used for convenience instead of ∠A ), and the side opposite to 90° – A is AB and the side adjacent to the angle 90°- A is BC as shown in the figure given above.

Comparing the above set of ratios with the ratios mentioned earlier, it can be seen that;

These relations are valid for all the values of A that lies between 0° and 90°.

19. Trigonometric Identities

Trigonometric Identities are useful whenever trigonometric functions are involved in an expression or an equation. Trigonometric Identities are true for every value of variables occurring on both sides of an equation. Geometrically, these identities involve certain trigonometric functions (such as sine, cosine, tangent) of one or more angles.

Sine, cosine and tangent are the primary trigonometry functions whereas cotangent, secant and cosecant are the other three functions. The trigonometric identities are based on all the six trig functions. Check Trigonometry Formulas to get formulas related to trigonometry.

Trigonometric Identities are useful whenever trigonometric functions are involved in an expression or an equation. Trigonometric Identities are true for every value of variables occurring on both sides of an equation. Geometrically, these identities involve certain trigonometric functions (such as sine, cosine, tangent) of one or more angles.

Sine, cosine and tangent are the primary trigonometry functions whereas cotangent, secant and cosecant are the other three functions. The trigonometric identities are based on all the six trig functions. Check Trigonometry Formulas to get formulas related to trigonometry.

Important Questions

Multiple Choice questions

1. If cos (α + β) = 0, then sin (α – β) can be reduced to

(a) cos β

(b) cos 2β

(c) sin α

(d) sin 2α

2. If cos (40° + A) = sin 30°, the value of A is:?

(a) 60°

(b) 20°

(c) 40°

(d) 30°

3. If sin x + cosec x = 2, then sin¹⁹x + cosec²⁰x =

(a) 2¹⁹

(b) 2²⁰

(c) 2

(d) 2³⁹

4. If cos 9a = sin a and 9a < 90°, then the value of tan 5a is

(a)

(b)

(c) 1

(d) 0

5. (1 + tanθ + secθ) (1 + cotθ – cosecθ) is equal to

(a) 0

(b) 1

(c) 2

(d) -1

6. Ratios of sides of a right triangle with respect to its acute angles are known as

(a) trigonometric identities

(b) trigonometry

(c) trigonometric ratios of the angles

(d) none of these

7. The value of cos θ cos(90° – θ) – sin θ sin (90° – θ) is:

(a) 1

(b) 0

(c) -1

(d) 2

8. If x = a cos θ and y = b sin θ, then b²x² + a²y² =

(a) ab

(b) b² + a²

(c) a²b²

(d) a⁴b⁴

9. If x and y are complementary angles, then

(a) sin x = sin y

(b) tan x = tan y

(c) cos x = cos y

(d) sec x = cosec y

10. sin (45° + θ) – cos (45° – θ) is equal to

(a) 2 cos θ

(b) 0

(c) 2 sin θ

(d) 1

Very Short Questions

1. Find maximum value of , 0°≤ θ ≤ 90°.

   

2. Given that sin θ = , find the value of tan θ.

   

3. If sin θ = cos θ, then find the value of 2 tan θ + cos² θ.
4. If sin (x – 20)° = cos (3x – 10)°, then find the value of x.
5. If sin² A = tan² 45°, where A is an acute angle, then find the value of A.

   

6. If x = a cos θ, y = b sin θ, then find the value of b²x² + a²y² – a²b².
7. If tan A = cot B, prove that A + B = 90°.
8. If sec A = 2x and tan A = 2x, find the value of 2(x²−) .

   

9. In a ∆ABC, if ∠C = 90°, prove that sin² A + sin² B = 1.
10. If sec 4A = cosec (A – 20°) where 4 A is an acute angle, find the value of A.

Short Questions

1. If sin A = , calculate cos A and tan A.

   

2. Given 15 cot A = 8, find sin A and sec A.
3. In Fig. 10.5, find tan P – cot R.

4. If sin θ + cos θ = √3 , then prove that tan θ + cot θ = 1.
5. Prove that = (sec θ – tan θ)²

   

6. 

7. 

8. 

9. Evaluate: sin 25° cos 65° + cos 25° sin 65°.
10. Without using tables, evaluate the following:

3 cos 68°. cosec 22° – tan 43°. tan 47°. tan 12°. tan 60°. tan 78°

Long Questions

1. In ∆PQR, right-angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P.
2. In triangle ABC right-angled at B, if tan A = find the value of:

   

(i) sin A cos C + cos A sin C (ii) cos A cos C – sin A sin C.

3. If cot θ = , evaluate:

   

(i)

(ii) cot² θ

4. If 3 cot A = 4, check whether = cos² A – sin² A or not.

   

5. Write all the other trigonometric ratios of ∠A in terms of sec A.
6. Prove that

7. Prove that:

8. Prove that:

9. Prove that: (sin θ + sec θ)² + (cos θ + cosec θ)² = (1 + sec θ cosec θ)².
10. Prove that:

Assertion Reason Questions

1. Two aeroplanes leave an airport, one after the other. After moving on runway, one flies due North and other flies due South. The speed of two aeroplanes is 400km/ hr and 500km/ hr respectively. Considering PQ as runway and A and B are any two points in the path followed by two planes, then answer the following questions.

2. Three friends - Anshu, Vijay and Vishal are playing hide and seek in a park. Anshu and Vijay hide in the shrubs and Vishal have to find both of them. If the positions of three friends are at A, B and C respectively as shown in the figure and forms a right angled triangle such that AB = 9 m, BC = and then answer the following questions.

   

Assertion Reason Questions

1. Directions: Each of these questions contains two statements: Assertion [A] and Reason [R]. Each of these questions also has four alternative choices, any one of which is the correct answer. You have to select one of the codes [a], [b], [c] and [d] given below.

1. A is true, R is true; R is a correct explanation for A.
2. A is true, R is true; R is not a correct explanation for A.
3. A is true; R is false.
4. A is false; R is true.

Assertion: The value of each of the trigonometric ratios of an angle does not depend on the size of the triangle. It only depends on the angle.

Reason: In right as hypotenuse is the longest side.

2. Directions: Each of these questions contains two statements: Assertion [A] and Reason [R]. Each of these questions also has four alternative choices, any one of which is the correct answer. You have to select one of the codes [a], [b], [c] and [d] given below.

1. A is true, R is true; R is a correct explanation for A.
2. A is true, R is true; R is not a correct explanation for A.
3. A is true; R is false.
4. A is false; R is true.

Assertion: Sin 60^o = Cos 30^o

Reason: Sin 2θ = Sin θ where θ is an acute angle.

Answer Key

Multiple Choice questions

1. (b) cos 2β
2. (b) 20°
3. (c) 2
4. (c) 1
5. (c) 2
6. (c) trigonometric ratios of the angles
7. (b) 0
8. (c) a²b²
9. (d) sec x = cosec y
10. (b) 0

Very Short Answer

1. , (0° ≤ θ ≤ 90°) (Given)

   

∵ sec θ is in the denominator

∴ The min. value of sec θ will return max. value for .

But the min. value of sec θ is sec 0° = 1.

Hence, the max. value of = = 1

2. sin θ =

   

3. sin θ = cos θ (Given)

It means value of θ = 45°

Now, 2 tan θ + cos² θ = 2 tan 45° + cos² 45°

4. sin (x – 20)° = cos (3x – 10)°

= 30°

5. sin²A = 12tan² 45°

⇒ sin2A = (1)² [∵ tan 45° = 1]

= sin2 A =

⇒ sin A =

Hence, ∠A = 45°

6. Given x = acos θ, y = b sin θ

b²x² + a²y² – a²b² = b²(acos θ)² + a²(b sin θ)² – a²b²

= a²b² cos²θ + a²b² sin² θ – a²b² = a²b² (sin² θ + cos² θ) – a²b²

= a²b² – a²b² = θ (∵ sin2 θ + cos² θ = 1)

7. We have

9. Since ∠C = 90°

∴ ∠A + ∠B = 180° – ∠C = 90°

Now, sin² A + sin² B = sin² A + sin² (90° – A) = sin² A + cos² A = 1

10. We have

∴ A = = 22°

Short Answer

1. Let us first draw a right ∆ABC in which ∠C = 90°.

Now, we know that

2. Let us first draw a right ∆ABC in which ∠B = 90°.

Now, we have, 15 cot A = 8

3. Using Pythagoras Theorem, we have

Now, tan P = and

tan P – cot R = – = 0

4. sin θ + cos θ = √3

⇒ 1 = tan θ + cot θ = 1

Therefore tan θ + cot θ = 1

9. sin 25°. cos 65° + cos 25° . sin 65°

10. We have,

3 cos 68°. cosec 22° – tan 43°. tan 47°. tan 12°. tan 60°. tan 78°.

= 3 cos (90° – 22°). cosec 22° – . {tan 43° . tan (90° – 43°)}. {tan 12°. tan (90° – 12°). tan 60°}

= 3 sin 22°. cosec 22° – (tan 43° . cot 43°). (tan 12°. cot 12°). tan 60°

= 3 × 1 – × 1 × 1 × √3

Long Answer

We have a right-angled ∆ABC in which ∠B = 90°.

and,

Now, tan A = = BCAB

3. Let us draw a right triangle ABC in which ∠B = 90° and ∠C = θ.

Let us consider a right triangle ABC in which ∠B = 90°

Let us consider a right-angled ∆ABC in which ∠B = 90°.

For ∠A we have

= (1 + sec θ cosec θ)² = RHS.

10. In order to show that,

Case Study Answers

1. Answer

2. Answer

Assertion Reason Answer

1. (b) A is true, R is true; R is not a correct explanation for A.
2. (c) A is true; R is false.