# NCERT Solutions | Class 11 Maths Chapter 3 | Trigonometric functions ## CBSE Solutions | Maths Class 11

Check the below NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric functions Pdf free download. NCERT Solutions Class 11 Maths  were prepared based on the latest exam pattern. We have Provided Trigonometric functions Class 11 Maths NCERT Solutions to help students understand the concept very well.

### NCERT | Class 11 Maths

Book: National Council of Educational Research and Training (NCERT) Central Board of Secondary Education (CBSE) 11th Maths 3 Trigonometric functions English

#### Trigonometric functions | Class 11 Maths | NCERT Books Solutions

You can refer to MCQ Questions for Class 11 Maths Chapter 3 Trigonometric functions to revise the concepts in the syllabus effectively and improve your chances of securing high marks in your board exams.

## NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.1

Ex 3.1 Class 11 Maths Question 1.
Find the radian measures corresponding to the following degree measures:
(i) 25°
(ii) -47°30′
(iii) 240°
(iv) 520°
Solution.
We have, 180° = π Radians Ex 3.1 Class 11 Maths Question 2.
Find the degree measures corresponding to the following radian measures $$\left( Use\quad \pi =\frac { 22 }{ 7 } \right)$$
(i) $$\frac { 11 }{ 16 }$$
(ii) -4
(iii) $$\frac { 5\pi }{ 3 }$$
(iv) $$\frac { 7\pi }{ 6 }$$
Solution.
We have π Radians = 180° Ex 3.1 Class 11 Maths Question 3.
A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second ?
Solution.
Number of revolutions made by wheel in one minute = 360
As we know that, 1 Revolution = 27 π Radians
∴ 360 Revolutions = 720 π Radians
∴ In 1 minute wheel can make = 720 π Radians
⇒ In 60 seconds wheel can make = 720 π Radians
⇒ In 1 second wheel can make
$$\frac { 720\pi }{ 3 } Radians=12\pi \quad Radians$$

Ex 3.1 Class 11 Maths Question 4.
Find the degree measure of the angle subtended at the centre of a circle of radius 100 cm by an arc of length 22 cm $$\left( Use\quad \pi =\frac { 22 }{ 7 } \right)$$
Solution.
Let O be the centre and AB be the arc length of the circle. Ex 3.1 Class 11 Maths Question 5.
In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of Xhe chord.
Solution.
Let AB be the minor arc of the chord.
AB = 20 cm, OA = OB = 20 cm Ex 3.1 Class 11 Maths Question 6.
If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their, radii.
Solution.
Let r1 r2 and θ1, θ2 be the radii and angles subtended at the centre of two circles respectively. Ex 3.1 Class 11 Maths Question 7.
Find the angle in radian through which a pendulum swings if its length is 75 cm and the tip describes an arc of length
(i) 10 cm
(ii) 15 cm
(iii) 21 cm
Solution. ## NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.2

Find the values of other five trigonometric functions in Exercises 1 to 5.

Ex 3.2 Class 11 Maths Question 1.
$$\cos { x } =\frac { -1 }{ 2 }$$, x lies in third quadrant.
Solution. Ex 3.2 Class 11 Maths Question 2.
$$\sin { x } =\frac { 3 }{ 5 }$$, x lies in second quadrant.
Solution. Ex 3.2 Class 11 Maths Question 3.
$$\cot { x= } \frac { 3 }{ 4 }$$, xlies in third quadrant.
Solution. Ex 3.2 Class 11 Maths Question 4.
$$\sec { x } =\frac { 13 }{ 5 }$$, x lies in fourth quadrant.
Solution. Ex 3.2 Class 11 Maths Question 5.
$$\tan { x } =-\frac { 5 }{ 12 }$$, x lies in second quadrant.
Solution. Find the values of the trigonometric functions in Exercises 6 to 10.

Ex 3.2 Class 11 Maths Question 6.
sin 765°
Solution. Ex 3.2 Class 11 Maths Question 7.
cosec (-1410°)
Solution. Ex 3.2 Class 11 Maths Question 8.
$$tan\quad \frac { 19\pi }{ 3 }$$
Solution. Ex 3.2 Class 11 Maths Question 9.
$$sin\left( -\frac { 11\pi }{ 3 } \right)$$
Solution. Ex 3.2 Class 11 Maths Question 10.
$$cot\left( -\frac { 15\pi }{ 4 } \right)$$
Solution. ## NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.3

Ex 3.3 Class 11 Maths Question 1.
Prove that: $${ sin }^{ 2 }\frac { \pi }{ 6 } +{ cos }^{ 2 }\frac { \pi }{ 3 } -{ tan }^{ 2 }\frac { \pi }{ 4 } =-\frac { 1 }{ 2 }$$
Solution.
L.H.S. = $${ sin }^{ 2 }\frac { \pi }{ 6 } +{ cos }^{ 2 }\frac { \pi }{ 3 } -{ tan }^{ 2 }\frac { \pi }{ 4 } =-\frac { 1 }{ 2 }$$
$$=\left[ { \left( \frac { 1 }{ 2 } \right) }^{ 2 }+{ \left( \frac { 1 }{ 2 } \right) }^{ 2 }-{ \left( 1 \right) }^{ 2 } \right] =\frac { 1 }{ 4 } +\frac { 1 }{ 4 } -1=\frac { -1 }{ 2 } =\quad R.H.S.$$

Ex 3.3 Class 11 Maths Question 2.
$$2{ sin }^{ 2 }\frac { \pi }{ 6 } +{ cosec }^{ 2 }\frac { 7\pi }{ 6 } { cos }^{ 2 }\frac { \pi }{ 3 } =\frac { 3 }{ 2 }$$
Solution.
L.H.S. = $$2{ sin }^{ 2 }\frac { \pi }{ 6 } +{ cosec }^{ 2 }\frac { 7\pi }{ 6 } { cos }^{ 2 }\frac { \pi }{ 3 }$$ Ex 3.3 Class 11 Maths Question 3.
$${ cot }^{ 2 }\frac { \pi }{ 6 } +cosec\frac { 5\pi }{ 6 } +3{ tan }^{ 2 }\frac { \pi }{ 6 } =6$$
Solution.
L.H.S. = $${ cot }^{ 2 }\frac { \pi }{ 6 } +cosec\frac { 5\pi }{ 6 } +3{ tan }^{ 2 }\frac { \pi }{ 6 }$$ Ex 3.3 Class 11 Maths Question 4.
$$2{ sin }^{ 2 }\frac { 3\pi }{ 4 } +2{ cos }^{ 2 }\frac { \pi }{ 4 } +2{ sec }^{ 2 }\frac { \pi }{ 3 } =10$$
Solution.
L.H.S. = $$2{ sin }^{ 2 }\frac { 3\pi }{ 4 } +2{ cos }^{ 2 }\frac { \pi }{ 4 } +2{ sec }^{ 2 }\frac { \pi }{ 3 }$$ Ex 3.3 Class 11 Maths Question 5.
Find the value of:
(i) sin 75°
(ii) tan 15°
Solution.
(i) sin (75°) = sin (30° + 45°) (ii) tan 15° = tan (45° – 30°) Prove the following:

Ex 3.3 Class 11 Maths Question 6.
$$cos\left( \frac { \pi }{ 4 } -x \right) cos\left( \frac { \pi }{ 4 } -y \right) -sin\left( \frac { \pi }{ 4 } -x \right) sin\left( \frac { \pi }{ 4 } -y \right)$$
Solution.
We have, Ex 3.3 Class 11 Maths Question 7.
$$\frac { tan\left( \frac { \pi }{ 4 } +x \right) }{ tan\left( \frac { \pi }{ 4 } -x \right) } ={ \left( \frac { 1+tan\quad x }{ 1-tan\quad x } \right) }^{ 2 }$$
Solution.
We have, Ex 3.3 Class 11 Maths Question 8.
$$\frac { cos\left( \pi +x \right) cos\left( -x \right) }{ sin\left( \pi -x \right) cos\left( \frac { \pi }{ 2 } +x \right) } ={ cot }^{ 2 }x$$
Solution.
We have, Ex 3.3 Class 11 Maths Question 9.
$$cos\left( \frac { 3\pi }{ 2 } +x \right) cos\left( 2\pi +x \right) \left[ cot\left( \frac { 3\pi }{ 2 } -x \right) +cot\left( 2\pi +x \right) \right] =1$$
Solution.
We have, Ex 3.3 Class 11 Maths Question 10.
sin(n +1 )x sin(n + 2)x + cos(n +1 )x cos(n + 2)x = cosx
Solution.
We have, Ex 3.3 Class 11 Maths Question 11.
$$cos\left( \frac { 3\pi }{ 4 } +x \right) -cos\left( \frac { 3\pi }{ 4 } -x \right) =-\sqrt { 2 } sinx$$
Solution.
We have, Ex 3.3 Class 11 Maths Question 12.
sin26x – sin24x= sin2x sin10x
Solution. Ex 3.3 Class 11 Maths Question 13.
cos22x – cos26x = sin 4x sin 8x
Solution. Ex 3.3 Class 11 Maths Question 14.
sin2x + 2 sin 4x + sin 6x = 4 cos2 x sin 4x
Solution.
We have, Ex 3.3 Class 11 Maths Question 15.
cot 4x (sin 5x + sin 3x) = cot x (sin 5x – sin 3x)
Solution. Ex 3.3 Class 11 Maths Question 16.
$$\frac { cos9x-cos5x }{ sin17x-sin3x } =-\frac { sin2x }{ cos10x }$$
Solution.
We have, Ex 3.3 Class 11 Maths Question 17.
$$\frac { sin5x+sin3x }{ cos5x+cos3x } =tan4x$$
Solution.
We have, Ex 3.3 Class 11 Maths Question 18.
$$\frac { sinx-siny }{ cosx+cosy } =tan\left( \frac { x-y }{ 2 } \right)$$
Solution. Ex 3.3 Class 11 Maths Question 19.
$$\frac { sinx+sin3x }{ cosx+cos3x } =tan2x$$
Solution. Ex 3.3 Class 11 Maths Question 20.
$$\frac { sinx-sin3x }{ { sin }^{ 2 }x-{ cos }^{ 2 }x } =2sinx$$
Solution. Ex 3.3 Class 11 Maths Question 21.
$$\frac { cos4x+cos3x+cos2x }{ sin4x+sin3x+sin2x } =cot3x$$
Solution. Ex 3.3 Class 11 Maths Question 22.
cot x cot 2x – cot 2x cot 3x – cot3x cotx = 1
Solution.
We know that 3x = 2x + x.
Therefore, Ex 3.3 Class 11 Maths Question 23.
$$tan4x=\frac { 4tanx\left( 1-{ tan }^{ 2 }x \right) }{ 1-6{ tan }^{ 2 }x+{ tan }^{ 4 }x }$$
Solution. Ex 3.3 Class 11 Maths Question 24.
cos 4x = 1 – 8 sin2x cos2x
Solution. Ex 3.3 Class 11 Maths Question 25.
cos 6x = 32 cos6 x – 48 cos4x + 18 cos2 x -1
Solution. ## NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.4

Find the principal and general solutions of the following equations:

Ex 3.4 Class 11 Maths Question 1.
$$tanx=\sqrt { 3 }$$
Solution. Ex 3.4 Class 11 Maths Question 2.
sec x = 2
Solution. Ex 3.4 Class 11 Maths Question 3.
$$cotx=-\sqrt { 3 }$$
Solution.  Ex 3.4 Class 11 Maths Question 4.
cosec x = -2
Solution. Find the general solution for each of the following equations:

Ex 3.4 Class 11 Maths Question 5.
cos 4x = cos 2x
Solution. Ex 3.4 Class 11 Maths Question 6.
cos 3x + cos x – cos 2x=0
Solution. Ex 3.4 Class 11 Maths Question 7.
sin 2 x + cos x = 0
Solution. Ex 3.4 Class 11 Maths Question 8.
sec22x = 1 – tan 2x
Solution. Ex 3.4 Class 11 Maths Question 9.
sin x + sin 3x + sin 5x = 0
Solution.  