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NCERT solutions Mathematics Class 10 chapter 8 Introduction to Trigonometry

Chapters

NCERT Mathematics Class 10

Mathematics Textbook for Class 10

Chapter 8 - Introduction to Trigonometry

Page 181

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

sin A, cos A

Q 1.1 | Page 181

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

Q 1.2 | Page 181

 In Given Figure, find tan P – cot R.

 

Q 2 | Page 181

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

Q 3 | Page 181

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

Q 4 | Page 181

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

Q 5 | Page 181

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

Q 6 | Page 181

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

Q 7.1 | Page 181

If cot θ = 7/8, evaluate cot2 θ

Q 7.2 | 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 8 | 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 9 | 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 10 | Page 181

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

The value of tan A is always less than 1.

Q 11.1 | Page 181

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

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

Q 11.2 | 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.

Q 11.3 | Page 181

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

Q 11.4 | Page 181

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

sin θ =4/3, for some angle θ

Q 11.5 | Page 181

Page 187

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

 

Q 1.1 | Page 187

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

Q 1.2 | Page 187

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

Q 1.3 | Page 187

Evaluate the following

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

Q 1.4 | Page 187

Evaluate the following

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

Q 1.5 | 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 2.1 | 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 2.2 | 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 2.3 | 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 2.4 | Page 187

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

Q 3 | Page 187

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

sin (A + B) = sin A + sin B

Q 4.1 | Page 187

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

The value of sinθ increases as θ increases

Q 4.2 | Page 187

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

The value of cos θ increases as θ increases

Q 4.3 | Page 187

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

sinθ = cos θ for all values of θ

Q 4.4 | Page 187

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

cot A is not defined for A = 0°

Q 4.5 | Page 187

Page 189

Evaluate `(sin 18^@)/(cos 72^@)`

Q 1.1 | Page 189

Evaluate `(tan 26^@)/(cot 64^@)`

 

Q 1.2 | Page 189

Evaluate cos 48° − sin 42°

Q 1.3 | Page 189

Evaluate cosec 31° − sec 59°

Q 1.4 | Page 189

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

Q 2.1 | Page 189

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

Q 2.2 | Page 189

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

Q 3 | Page 189

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

Q 4 | Page 189

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

Q 5 | 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 6 | Page 189

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

Q 7 | Page 189

Pages 193 - 194

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

Q 1 | Page 193

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

Q 2 | Page 193
 

Evaluate

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

 
Q 3.1 | Page 193

 Evaluate sin25° cos65° + cos25° sin65°

Q 3.2 | Page 193

Choose the correct option. Justify your choice.

9 sec2 A − 9 tan2 A =

(A) 1

(B) 9

(C) 8

(D) 0

Q 4.1 | 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 4.2 | Page 193

Choose the correct option. Justify your choice.

(secA + tanA) (1 − sinA) =

A) secA

(B) sinA

(C) cosecA

(D) cosA

Q 4.3 | Page 193

Choose the correct option. Justify your choice.

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

A) secA

(B) −1

(C) cotA

(D) tanA

Q 4.4 | 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 5.1 | 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 5.2 | Page 193

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 5.3 | 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 5.4 | 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 5.5 | 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 5.6 | 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`

Q 5.7 | Page 194

Extra questions

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

Evaluate the following 

sec 50º sin 40° + cos 40º cosec 50º 

Find the value of x in each of the following :

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

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

Prove the following trigonometric identities:

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

Prove the following trigonometric identities:

(i) (1 – sin2θ) sec2θ = 1

(ii) cos2θ (1 + tan2θ) = 1

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

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 }`

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

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

Without using trigonometric tables evaluate the following:

`(i) sin^2 25º + sin^2 65º `

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.`

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

 

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.`

Find the value of x in the following :

tan 3x = sin 45º cos 45º + sin 30º

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

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

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

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

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

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,

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}`

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

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

Find the value of θ in each of the following :

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

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}}`

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`

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.

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

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

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

 

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

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 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 } )`

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`

NCERT Mathematics Class 10

Mathematics Textbook for Class 10
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