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

## Chapter 8 : Introduction to Trigonometry

#### Page 181

Q 1.1 | Page 181

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

sin A, cos A

Q 1.2 | Page 181

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

Q 2 | Page 181

In Given Figure, find tan P – cot R.

Q 3 | Page 181

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

Q 4 | Page 181

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

Q 5 | Page 181

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

Q 6 | Page 181

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

Q 7.1 | Page 181

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

Q 7.2 | Page 181

If cot θ = 7/8, evaluate cot2 θ

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

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

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.

Q 11.1 | Page 181

The value of tan A is always less than 1.

Q 11.2 | Page 181

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

Q 11.3 | Page 181

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

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

Q 11.5 | Page 181

sin θ =4/3, for some angle θ

#### Page 187

Q 1.1 | Page 187

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

Q 1.2 | Page 187

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

Q 1.3 | Page 187

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

Q 1.4 | Page 187

Evaluate the following

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

Q 1.5 | Page 187

Evaluate the following

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

Q 2.1 | Page 187

Choose the correct option and justify your choice

(2 tan 30°)/(1+tan^2 30°)

(A) sin 60°

(B) cos 60°

(C) tan 60°

(D) sin 30°

Q 2.2 | Page 187

Choose the correct option and justify your choice.

(1- tan^2 45°)/(1+tan^2 45°)

(A) tan 90°

(B) 1

(C) sin 45°

(D) 0

Q 2.3 | Page 187

Choose the correct option and justify your choice :

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

(A) 0°

(B) 30°

(C) 45°

(D) 60°

Q 2.4 | Page 187

Choose the correct option and justify your choice :

(2 tan 30°)/(1-tan^2 30°)

(A) cos 60°

(B) sin 60°

(C) tan 60°

(D) sin 30°

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.

Q 4.1 | Page 187

sin (A + B) = sin A + sin B

Q 4.2 | Page 187

The value of sinθ increases as θ increases

Q 4.3 | Page 187

The value of cos θ increases as θ increases

Q 4.4 | Page 187

sinθ = cos θ for all values of θ

Q 4.5 | Page 187

cot A is not defined for A = 0°

#### Page 189

Q 1.1 | Page 189

Evaluate (sin 18^@)/(cos 72^@)

Q 1.2 | Page 189

Evaluate (tan 26^@)/(cot 64^@)

Q 1.3 | Page 189

Evaluate cos 48° − sin 42°

Q 1.4 | Page 189

Evaluate cosec 31° − sec 59°

Q 2.1 | Page 189

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

Q 2.2 | Page 189

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

Q 3 | Page 189

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

Q 4 | Page 189

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

Q 5 | Page 189

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

Q 6 | Page 189

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

Q 7 | Page 189

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

#### Pages 193 - 194

Q 1 | Page 193

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

Q 2 | Page 193

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

Q 3.1 | Page 193

Evaluate

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

Q 3.2 | Page 193

Evaluate sin25° cos65° + cos25° sin65°

Q 4.1 | Page 193

Choose the correct option. Justify your choice.

9 sec2 A − 9 tan2 A =

(A) 1

(B) 9

(C) 8

(D) 0

Q 4.2 | Page 193

Choose the correct option. Justify your choice.

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

(A) 0

(B) 1

(C) 2

(D) −1

Q 4.3 | Page 193

Choose the correct option. Justify your choice.

(secA + tanA) (1 − sinA) =

A) secA

(B) sinA

(C) cosecA

(D) cosA

Q 4.4 | Page 193

Choose the correct option. Justify your choice.

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

A) secA

(B) −1

(C) cotA

(D) tanA

Q 5.1 | 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)

Q 5.2 | 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

Q 5.3 | 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

Q 5.4 | 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]

Q 5.5 | 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

Q 5.6 | 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

Q 5.7 | 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

#### Extra questions

Without using trigonometric tables, evaluate the following:

( i)\frac{\cos37^\text{o}}{\sin53^\text{o}}\text{ }(ii)\frac{\sin41^\text{o}}{\cos 49^\text{o}}(iii)\frac{\sin30^\text{o}17'}{\cos59^\text{o}\43'}

If θ is an acute angle and sin θ = cos θ, find the value of 2 tan2 θ + sin2 θ – 1

Prove that: (1 – sinθ + cosθ)^2 = 2(1 + cosθ)(1 – sinθ)

Prove the following identities:

(i) 2 (sin^6 θ + cos^6 θ) –3(sin^4 θ + cos^4 θ) + 1 = 0

(ii) (sin^8 θ – cos^8 θ) = (sin^2 θ – cos^2 θ) (1 – 2sin^2 θ cos^2 θ)

Prove the following trigonometric identities:

(i) (1 – sin2θ) sec2θ = 1

(ii) cos2θ (1 + tan2θ) = 1

Prove the following identities:

(i) cos4^4 A – cos^2 A = sin^4 A – sin^2 A

(ii) cot^4 A – 1 = cosec^4 A – 2cosec^2 A

(iii) sin^6 A + cos^6 A = 1 – 3sin^2 A cos^2 A.

If secθ + tanθ = p, show that (p^{2}-1)/(p^{2}+1)=\sin \theta

An equilateral triangle is inscribed in a circle of radius 6 cm. Find its side.

Prove the following trigonometric identities:

(\text{i})\text{ }\frac{\sin \theta }{1-\cos \theta }=\text{cosec}\theta+\cot \theta

If sinθ + cosθ = p and secθ + cosecθ = q, show that q(p2 – 1) = 2p

Evaluate the following expression:

(i) tan 60º cosec^2 45º + sec^2 60º tan 45º

(ii) 4cot^2 45º – sec^2 60º + sin^2 60º + cos^2 90º.

"If "\frac{\cos \alpha }{\cos \beta }=m\text{ and }\frac{\cos \alpha }{\sin \beta }=n " show that " (m^2 + n^2 ) cos^2 β = n^2

If tanθ + sinθ = m and tanθ – sinθ = n, show that m^2 – n^2 = 4\sqrt{mn}.

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

If m=(acosθ + bsinθ) and n=(asinθ – bcosθ) prove that m2+n2=a2+b2

Prove that \frac{\sin \theta -\cos \theta }{\sin \theta +\cos \theta }+\frac{\sin\theta +\cos \theta }{\sin \theta -\cos \theta }=\frac{2}{2\sin^{2}\theta -1}

Find the value of x in the following :

tan 3x = sin 45º cos 45º + sin 30º

Evaluate the following in the simplest form: sin 60º cos 45º + cos 60º sin 45º

If acosθ – bsinθ = c, prove that asinθ + bcosθ = \pm \sqrt{a^{2}+b^{2}-c^{2}

Prove that  \frac{\sin \theta -\cos \theta +1}{\sin\theta +\cos \theta -1}=\frac{1}{\sec \theta -\tan \theta } using the identity sec2 θ = 1 + tan2 θ.

Using the formula, sin(A – B) = sinA cosB – cosA sinB, find the value of sin 15º

If sinθ + sin2 θ = 1, prove that cos2 θ + cos4 θ = 1

Find the value of θ in each of the following :

(i) 2 sin 2θ = √3      (ii) 2 cos 3θ = 1

Evaluate the following

sec 50º sin 40° + cos 40º cosec 50º

Prove the following identities:

( i)sin^{2}A/cos^{2}A+\cos^{2}A/sin^{2}A=\frac{1}{sin^{2}Acos^{2}A)-2

(ii)\frac{cosA}{1tanA}+\sin^{2}A/(sinAcosA)=\sin A\text{}+\cos A

( iii)((1+sin\theta )^{2}+(1sin\theta)^{2})/cos^{2}\theta =2( \frac{1+sin^{2}\theta}{1-sin^{2}\theta } )

Show that:

(i) 2(cos^2 45º + tan^2 60º) – 6(sin^2 45º – tan^2 30º) = 6

(ii) 2(cos^4 60º + sin^4 30º) – (tan^2 60º + cot^2 45º) + 3 sec^2 30º = 1/4

If x = 30°, verify that

(i) \tan 2x=\frac{2\tan x}{1-\tan ^{2}x

(ii) \sin x=\sqrt{\frac{1-\cos 2x}{2}}

If (secA + tanA)(secB + tanB)(secC + tanC) = (secA – tanA)(secB – tanB)(secC – tanC) prove that each of the side is equal to ±1. We have,

(\text{i})\text{ }\frac{\cot 54^\text{o}}{\tan36^\text{o}}+\frac{\tan 20^\text{o}}{\cot 70^\text{o}}-2

If cosθ + sinθ = √2 cosθ, show that cosθ – sinθ = √2 sinθ.

Prove the following identities:

(i) (sinθ + cosecθ)^2 + (cosθ + secθ)^2 = 7 + tan^2 θ + cot^2 θ

(ii) (sinθ + secθ)^2 + (cosθ + cosecθ)^2 = (1 + secθ cosecθ)^2

(iii) sec^4 θ– sec^2 θ = tan^4 θ + tan^2 θ

Find the value of x in each of the following :

cos x = cos 60º cos 30º + sin 60º sin 30º

\text{Evaluate }\frac{\tan 65^\circ }{\cot 25^\circ}

Without using trigonometric tables evaluate the following:

(i) sin^2 25º + sin^2 65º

Express each of the following in terms of trigonometric ratios of angles between 0º and 45º;

(i) cosec 69º + cot 69º

(ii) sin 81º + tan 81º

(iii) sin 72º + cot 72º

Without using trigonometric tables, evaluate the following:

(\sin ^{2}20^\text{o}+\sin^{2}70^\text{o})/(\cos ^{2}20^\text{o}+\cos ^{2}70^\text{o}}+\frac{\sin (90^\text{o}-\theta )\sin \theta }{\tan \theta }+\frac{\cos (90^\text{o}-\theta )\cos \theta }{\cot \theta }

If A, B, C are the interior angles of a triangle ABC, prove that \tan \frac{B+C}{2}=\cot \frac{A}{2}

If tan 2θ = cot (θ + 6º), where 2θ and θ + 6º are acute angles, find the value of θ

## 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 Trigonometric Identities, Trigonometric Ratios of Complementary Angles, Trigonometric Ratios, Relationships Between the Ratios, Trigonometric Ratios of Some Specific Angles, Proof of Existence, Trigonometric Ratios of an Acute Angle of a Right-angled Triangle, Introduction to Trigonometry, Introduction to Trigonometry Examples and Solutions.

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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