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NCERT solutions for Class 10 Mathematics chapter 8 - Introduction to Trigonometry

Mathematics Textbook for Class 10

NCERT Mathematics Class 10 Chapter 8: Introduction to Trigonometry

Ex. 8.10Ex. 8.20Ex. 8.30Ex. 8.40

Chapter 8: Introduction to Trigonometry Exercise 8.10 solutions [Page 181]

Ex. 8.10 | Q 1.1 | Page 181

In ΔABC right angled at B, AB = 24 cm, BC = 7 m. Determine

sin A, cos A

Ex. 8.10 | Q 1.2 | Page 181

In ΔABC right angled at B, AB = 24 cm, BC = 7 m. Determine sin C, cos C

Ex. 8.10 | Q 2 | Page 181

In Given Figure, find tan P – cot R. Ex. 8.10 | Q 3 | Page 181

If sin A =3/4 , calculate cos A and tan A.

Ex. 8.10 | Q 4 | Page 181

Given 15 cot A = 8. Find sin A and sec A

Ex. 8.10 | Q 5 | Page 181

Given sec θ = 13/12 , calculate all other trigonometric ratios.

Ex. 8.10 | Q 6 | Page 181

If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B.

Ex. 8.10 | Q 7.1 | Page 181

If cot θ = 7/8 evaluate ((1+sin θ )(1-sin θ))/((1+cos θ)(1-cos θ))

Ex. 8.10 | Q 7.2 | Page 181

If cot θ = 7/8, evaluate cot2 θ

Ex. 8.10 | Q 8 | Page 181

If 3 cot A = 4, Check whether ((1-tan^2 A)/(1+tan^2 A)) = cos^2 A - sin^2 A or not

Ex. 8.10 | Q 9 | Page 181

In ΔABC, right angled at B. If tan A = 1/sqrt3 , find the value of

(i) sin A cos C + cos A sin C

(ii) cos A cos C − sin A sin C

Ex. 8.10 | Q 10 | Page 181

In ΔPQR, right angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P.

Ex. 8.10 | Q 11.1 | Page 181

State whether the following are true or false. Justify your answer.

The value of tan A is always less than 1.

Ex. 8.10 | Q 11.2 | Page 181

State whether the following are true or false. Justify your answer.

sec A = 12/5 for some value of angle A.

Ex. 8.10 | Q 11.3 | Page 181

State whether the following are true or false. Justify your answer.

cos A is the abbreviation used for the cosecant of angle A.

Ex. 8.10 | Q 11.4 | Page 181

State whether the following are true or false. Justify your answer. cot A is the product of cot and A

Ex. 8.10 | Q 11.5 | Page 181

State whether the following are true or false. Justify your answer.

sin θ =4/3, for some angle θ

Chapter 8: Introduction to Trigonometry Exercise 8.20 solutions [Page 187]

Ex. 8.20 | Q 1.1 | Page 187

Evaluate the following in the simplest form: sin 60º cos 30º + cos 60º sin 30º

Ex. 8.20 | Q 1.2 | Page 187

Evaluate the following : 2tan245° + cos230° − sin260°

Ex. 8.20 | Q 1.3 | Page 187

Evaluate the following : (cos 45°)/(sec 30° + cosec 30°)

Ex. 8.20 | Q 1.4 | Page 187

Evaluate the following

(sin 30° + tan 45° – cosec 60°)/(sec 30° + cos 60° + cot 45°)

Ex. 8.20 | Q 1.5 | Page 187

Evaluate the following

(5cos^2 60° + 4sec^2 30° - tan^2 45°)/(sin^2 30°+cos^2 30°)

Ex. 8.20 | Q 2.1 | Page 187

Choose the correct option and justify your choice

(2 tan 30°)/(1+tan^2 30°)

• sin 60°

• cos 60°

• tan 60°

• sin 30°

Ex. 8.20 | Q 2.2 | Page 187

Choose the correct option and justify your choice.

(1- tan^2 45°)/(1+tan^2 45°)

• tan 90°

• 1

• sin 45°

• 0

Ex. 8.20 | Q 2.3 | Page 187

Choose the correct option and justify your choice :

sin 2A = 2 sin A is true when A =

• 30°

• 45°

• 60°

Ex. 8.20 | Q 2.4 | Page 187

Choose the correct option and justify your choice :

(2 tan 30°)/(1-tan^2 30°)

• cos 60°

• sin 60°

• tan 60°

• sin 30°

Ex. 8.20 | Q 3 | Page 187

If tan (A + B) = sqrt3 and tan (A – B) = 1/sqrt3 ; 0° < A + B ≤ 90° ; A > B, find A and B.

Ex. 8.20 | Q 4.1 | Page 187

State whether the following is true or false. Justify your answer.

sin (A + B) = sin A + sin B

• True

• False

Ex. 8.20 | Q 4.2 | Page 187

State whether the following is true or false. Justify your answer.

The value of sinθ increases as θ increases

• True

• False

Ex. 8.20 | Q 4.3 | Page 187

State whether the following is true or false. Justify your answer.

The value of cos θ increases as θ increases

• True

• False

Ex. 8.20 | Q 4.4 | Page 187

State whether the following is true or false. Justify your answer

sinθ = cos θ for all values of θ

• True

• False

Ex. 8.20 | Q 4.5 | Page 187

State whether the following are true or false. Justify your answer.

cot A is not defined for A = 0°

• True

• False

Chapter 8: Introduction to Trigonometry Exercise 8.30 solutions [Pages 189 - 190]

Ex. 8.30 | Q 1.1 | Page 189

Evaluate (sin 18^@)/(cos 72^@)

Ex. 8.30 | Q 1.1 | Page 189

Evaluate (sin 18^@)/(cos 72^@)

Ex. 8.30 | Q 1.2 | Page 189

Evaluate (tan 26^@)/(cot 64^@)

Ex. 8.30 | Q 1.2 | Page 189

Evaluate (tan 26^@)/(cot 64^@)

Ex. 8.30 | Q 1.3 | Page 189

Evaluate cos 48° − sin 42°

Ex. 8.30 | Q 1.4 | Page 189

Evaluate cosec 31° − sec 59°

Ex. 8.30 | Q 1.4 | Page 189

Evaluate cosec 31° − sec 59°

Ex. 8.30 | Q 2.1 | Page 189

Show that tan 48° tan 23° tan 42° tan 67° = 1

Ex. 8.30 | Q 2.2 | Page 189

Show that cos 38° cos 52° − sin 38° sin 52° = 0

Ex. 8.30 | Q 2.2 | Page 189

Show that cos 38° cos 52° − sin 38° sin 52° = 0

Ex. 8.30 | Q 3 | Page 189

If tan 2A = cot (A – 18°), where 2A is an acute angle, find the value of A

Ex. 8.30 | Q 3 | Page 189

If tan 2A = cot (A – 18°), where 2A is an acute angle, find the value of A

Ex. 8.30 | Q 4 | Page 189

If tan A = cot B, prove that A + B = 90

Ex. 8.30 | Q 4 | Page 189

If tan A = cot B, prove that A + B = 90

Ex. 8.30 | Q 5 | Page 189

If sec 4A = cosec (A− 20°), where 4A is an acute angle, find the value of A.

Ex. 8.30 | Q 5 | Page 189

If sec 4A = cosec (A− 20°), where 4A is an acute angle, find the value of A.

Ex. 8.30 | Q 6 | Page 190

If A, B and C are interior angles of a triangle ABC, then show that \sin( \frac{B+C}{2} )=\cos \frac{A}{2}

Ex. 8.30 | Q 7 | Page 190

Express sin 67° + cos 75° in terms of trigonometric ratios of angles between 0° and 45°

Ex. 8.30 | Q 7 | Page 190

Express sin 67° + cos 75° in terms of trigonometric ratios of angles between 0° and 45°

Chapter 8: Introduction to Trigonometry Exercise 8.40 solutions [Pages 193 - 194]

Ex. 8.40 | Q 1 | Page 193

Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.

Ex. 8.40 | Q 1 | Page 193

Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.

Ex. 8.40 | Q 2 | Page 193

Write all the other trigonometric ratios of ∠A in terms of sec A.

Ex. 8.40 | Q 2 | Page 193

Write all the other trigonometric ratios of ∠A in terms of sec A.

Ex. 8.40 | Q 3.1 | Page 193

Evaluate

(sin ^2 63^@ + sin^2 27^@)/(cos^2 17^@+cos^2 73^@)

Ex. 8.40 | Q 3.1 | Page 193

Evaluate

(sin ^2 63^@ + sin^2 27^@)/(cos^2 17^@+cos^2 73^@)

Ex. 8.40 | Q 3.2 | Page 193

Evaluate sin25° cos65° + cos25° sin65°

Ex. 8.40 | Q 3.2 | Page 193

Evaluate sin25° cos65° + cos25° sin65°

Ex. 8.40 | Q 4.1 | Page 193

Choose the correct option. Justify your choice.

9 sec2 A − 9 tan2 A =

• 1

• 9

• 8

• 0

Ex. 8.40 | Q 4.1 | Page 193

Choose the correct option. Justify your choice.

9 sec2 A − 9 tan2 A =

• 1

• 9

• 8

• 0

Ex. 8.40 | Q 4.2 | Page 193

Choose the correct option. Justify your choice.

(1 + tan θ + sec θ) (1 + cot θ − cosec θ)

• 0

• 1

• 2

• -1

Ex. 8.40 | Q 4.2 | Page 193

Choose the correct option. Justify your choice.

(1 + tan θ + sec θ) (1 + cot θ − cosec θ)

• 0

• 1

• 2

• -1

Ex. 8.40 | Q 4.3 | Page 193

Choose the correct option. Justify your choice.

(secA + tanA) (1 − sinA) =

• secA

• sinA

• cosecA

• cosA

Ex. 8.40 | Q 4.3 | Page 193

Choose the correct option. Justify your choice.

(secA + tanA) (1 − sinA) =

• secA

• sinA

• cosecA

• cosA

Ex. 8.40 | Q 4.4 | Page 193

Choose the correct option. Justify your choice.

(1+tan^2A)/(1+cot^2A)

• secA

• −1

• cotA

• tanA

Ex. 8.40 | Q 4.4 | Page 193

Choose the correct option. Justify your choice.

(1+tan^2A)/(1+cot^2A)

• secA

• −1

• cotA

• tanA

Ex. 8.40 | Q 5.01 | Page 193

Prove the following identities, where the angles involved are acute angles for which the expressions are defined

(cosec θ – cot θ)^2 = (1-cos theta)/(1 + cos theta)

Ex. 8.40 | Q 5.01 | Page 193

Prove the following identities, where the angles involved are acute angles for which the expressions are defined

(cosec θ – cot θ)^2 = (1-cos theta)/(1 + cos theta)

Ex. 8.40 | Q 5.02 | Page 193

Prove the following identities, where the angles involved are acute angles for which the expressions are defined

cos A/(1 + sin A) + (1 + sin A)/cos A = 2 sec A

Ex. 8.40 | Q 5.02 | Page 193

Prove the following identities, where the angles involved are acute angles for which the expressions are defined

cos A/(1 + sin A) + (1 + sin A)/cos A = 2 sec A

Ex. 8.40 | Q 5.03 | Page 194

Prove the following identities, where the angles involved are acute angles for which the expressions are defined

(tantheta)/(1-cottheta) + (cottheta)/(1-tantheta) = 1+secthetacosectheta

Ex. 8.40 | Q 5.03 | Page 194

Prove the following identities, where the angles involved are acute angles for which the expressions are defined

(tantheta)/(1-cottheta) + (cottheta)/(1-tantheta) = 1+secthetacosectheta

Ex. 8.40 | Q 5.04 | Page 194

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

(1+ secA)/sec A = (sin^2A)/(1-cosA)

[Hint : Simplify LHS and RHS separately]

Ex. 8.40 | Q 5.04 | Page 194

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

(1+ secA)/sec A = (sin^2A)/(1-cosA)

[Hint : Simplify LHS and RHS separately]

Ex. 8.40 | Q 5.05 | Page 194

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

(cos A-sinA+1)/(cosA+sinA-1)=cosecA+cotA

Ex. 8.40 | Q 5.05 | Page 194

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

(cos A-sinA+1)/(cosA+sinA-1)=cosecA+cotA

Ex. 8.40 | Q 5.06 | Page 194

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

sqrt((1+sinA)/(1-sinA)) = secA + tanA

Ex. 8.40 | Q 5.06 | Page 194

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

sqrt((1+sinA)/(1-sinA)) = secA + tanA

Ex. 8.40 | Q 5.07 | Page 194

Prove the following identities, where the angles involved are acute angles for which the expressions are defined

(sin theta-2sin^3theta)/(2cos^3theta -costheta) = tan theta

Ex. 8.40 | Q 5.07 | Page 194

Prove the following identities, where the angles involved are acute angles for which the expressions are defined

(sin theta-2sin^3theta)/(2cos^3theta -costheta) = tan theta

Ex. 8.40 | Q 5.08 | Page 194

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
(sin A + cosec A)2 + (cos A + sec A)2 = 7 + tan2 A + cot2 A

Ex. 8.40 | Q 5.09 | Page 194

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
(cosec A – sin A) (sec A – cos A)=1/(tanA+cotA)

[Hint : Simplify LHS and RHS separately]

Ex. 8.40 | Q 5.1 | Page 194

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

((1+tan^2A)/(1+cot^2A))=((1-tanA)/(1-cotA))^2=tan^2A

Chapter 8: Introduction to Trigonometry

Ex. 8.10Ex. 8.20Ex. 8.30Ex. 8.40

NCERT Mathematics Class 10 NCERT solutions for Class 10 Mathematics chapter 8 - Introduction to Trigonometry

NCERT solutions for Class 10 Maths chapter 8 (Introduction to Trigonometry) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Textbook for Class 10 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com are providing such solutions so that students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 10 Mathematics chapter 8 Introduction to Trigonometry are Introduction to Trigonometry, Introduction to Trigonometry Examples and Solutions, Trigonometric Ratios, Trigonometric Ratios of an Acute Angle of a Right-angled Triangle, Trigonometric Ratios of Some Specific Angles, Trigonometric Ratios of Complementary Angles, Trigonometric Identities, Proof of Existence, Relationships Between the Ratios, Trigonometric Ratios of Complementary Angles, Trigonometric Identities.

Using NCERT Class 10 solutions Introduction to Trigonometry exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in NCERT Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 10 prefer NCERT Textbook Solutions to score more in exam.

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