PYQs Guides Notes Mocks Board Questions Maharashtra HSC Class 12 Mathematics Trigonometric Functions 50 Maharashtra HSC Class 12 Mathematics previous year questions (PYQs) from 14 papers, 2015 to 2026, with answers and worked solutions.
About this chapter Difficulty: 16 easy · 34 moderate. Most-asked subtopics: Properties of Inverse Trigonometric Functions (13), Trigonometric Equations and General Solutions (12), Solution of Triangle — Sine, Cosine and Projection Rules (10). Updated 19 September 2026. #1 · Trigonometric Equations and General Solutions
Find the general solution of equation cos 5 θ = 1 2 \cos 5\theta = \frac{1}{2} cos 5 θ = 2 1 . #2 · Trigonometric Equations and General Solutions
Find the general solution of tan θ = 0 \tan \theta = 0 tan θ = 0 . #3 · Trigonometric Equations and General Solutions
Find the general solution of 4 cos 2 θ = 3 4\cos^{2}\theta = 3 4 cos 2 θ = 3 . Maharashtra HSC Class 12 · 2025 · Q. 5 Easy Add #4 · Trigonometric Equations and General Solutions
Find the general solution of tan 2 θ = 1 \tan^{2}\theta = 1 tan 2 θ = 1 . Maharashtra HSC Class 12 · 2024 · Q. 1. (ii) Easy Add #5 · Trigonometric Equations and General Solutions
The principle solutions of the equation c o s θ = 1 2 cos\theta = \frac{1}{2} cos θ = 2 1 are _______. A π 6 , 5 π 6 \frac{\pi}{6},\frac{5\pi}{6} 6 π , 6 5 π B π 3 , 5 π 3 \frac{\pi}{3},\frac{5\pi}{3} 3 π , 3 5 π C π 6 , 7 π 6 \frac{\pi}{6},\frac{7\pi}{6} 6 π , 6 7 π D π 3 , 2 π 3 \frac{\pi}{3},\frac{2\pi}{3} 3 π , 3 2 π Done a few? Here’s your next move
Maharashtra HSC Class 12 · 2023 · Q. 15 Moderate Add #6 · Trigonometric Equations and General Solutions
Find the general solution of s i n θ + s i n 3 θ + s i n 5 θ = 0 sin\theta + sin3\theta + sin5\theta = 0 s in θ + s in 3 θ + s in 5 θ = 0 . Maharashtra HSC Class 12 · 2022 · Q. 5 Easy Add #7 · Trigonometric Equations and General Solutions
Find the principal solutions of c o t θ = 0 cot\theta = 0 co tθ = 0 . Maharashtra HSC Class 12 · 2019 · Q. 1. Easy Add #8 · Trigonometric Equations and General Solutions
The principal solutions of c o t x = − 3 cotx = - \sqrt{3} co t x = − 3 are A π 6 , 7 π 6 \frac{\pi}{6},\frac{7\pi}{6} 6 π , 6 7 π B 5 π 6 , 7 π 6 \frac{5\pi}{6},\frac{7\pi}{6} 6 5 π , 6 7 π C 5 π 6 , 11 π 6 \frac{5\pi}{6},\frac{11\pi}{6} 6 5 π , 6 11 π D π 6 , 11 π 6 \frac{\pi}{6},\frac{11\pi}{6} 6 π , 6 11 π Maharashtra HSC Class 12 · 2018 · Q. 1. ii. Easy Add #9 · Trigonometric Equations and General Solutions
The principal solutions of s e c x = 2 3 secx = \frac{2}{\sqrt{3}} sec x = 3 2 are A π 3 , 11 π 6 \frac{\pi}{3},\frac{11\pi}{6} 3 π , 6 11 π B π 6 , 11 π 6 \frac{\pi}{6},\frac{11\pi}{6} 6 π , 6 11 π C π 4 , 11 π 4 \frac{\pi}{4},\frac{11\pi}{4} 4 π , 4 11 π D π 6 , 11 π 4 \frac{\pi}{6},\frac{11\pi}{4} 6 π , 4 11 π Maharashtra HSC Class 12 · 2017 · Q. 2. (A) iii. Moderate Add #10 · Trigonometric Equations and General Solutions
Find the general solution of s i n 2 x + s i n 4 x + s i n 6 x = 0 sin2x + sin4x + sin6x = 0 s in 2 x + s in 4 x + s in 6 x = 0 . Maharashtra HSC Class 12 · 2016 · Q. 3. (B) iii. Moderate Add #11 · Trigonometric Equations and General Solutions
Find the general solution of s i n x + s i n 3 x + s i n 5 x = 0 sinx + sin3x + sin5x = 0 s in x + s in 3 x + s in 5 x = 0 . Maharashtra HSC Class 12 · 2015 · Q. 3. (B) i. Moderate Add #12 · Trigonometric Equations and General Solutions
Find the general solution of c o s x − s i n x = 1. cosx - sinx = 1. cos x − s in x = 1. #13 · Polar Coordinates
Find the polar co-ordinates of the point whose Cartesian co-ordinates are ( 1 , − 3 ) \left(1, -\sqrt{3}\right) ( 1 , − 3 ) . #14 · Polar Coordinates
Find the polar coordinates of the point whose cartesian coordinates are ( − 2 , − 2 ) (-\sqrt{2}, -\sqrt{2}) ( − 2 , − 2 ) . #15 · Solution of Triangle — Sine, Cosine and Projection Rules
In Δ A B C \Delta ABC Δ A B C , if a = 10 a = 10 a = 10 , b = 8 b = 8 b = 8 , c = 6 c = 6 c = 6 then find the value of
(i) sin ( C 2 ) \sin\left(\frac{C}{2}\right) sin ( 2 C ) (ii) cos ( B 2 ) \cos\left(\frac{B}{2}\right) cos ( 2 B ) #16 · Solution of Triangle — Sine, Cosine and Projection Rules
In Δ A B C \Delta ABC Δ A B C , if a cos A = b cos B a \cos A = b \cos B a cos A = b cos B then prove that the triangle is either a right angled or an isosceles triangle. #17 · Solution of Triangle — Sine, Cosine and Projection Rules
In Δ A B C \Delta ABC Δ A B C , if a = 2 , b = 3 a = 2, b = 3 a = 2 , b = 3 and sin A = 2 3 \sin A = \frac{2}{3} sin A = 3 2 , then ∠ B = \angle B = ∠ B = _____ #18 · Solution of Triangle — Sine, Cosine and Projection Rules
In Δ A B C \Delta ABC Δ A B C , if a = 2 , b = 3 , c = 4 a = 2, b = 3, c = 4 a = 2 , b = 3 , c = 4 , then prove that the triangle is obtuse angled. #19 · Solution of Triangle — Sine, Cosine and Projection Rules
In Δ A B C \Delta ABC Δ A B C , prove that a 2 = b 2 + c 2 − 2 b c cos A a^{2} = b^{2} + c^{2} - 2bc \cos A a 2 = b 2 + c 2 − 2 b c cos A #20 · Solution of Triangle — Sine, Cosine and Projection Rules
In Δ A B C \Delta ABC Δ A B C , prove that a ( b cos C − c cos B ) = b 2 − c 2 a(b\cos C - c\cos B) = b^{2} - c^{2} a ( b cos C − c cos B ) = b 2 − c 2 . Maharashtra HSC Class 12 · 2025 · Q. 1. ii. Easy Add #21 · Solution of Triangle — Sine, Cosine and Projection Rules
In △ A B C \bigtriangleup ABC △ A B C , ( a + b ) c o s C + ( b + c ) c o s A + ( c + a ) c o s B (a + b)cosC + (b + c)cosA + (c + a)cosB ( a + b ) cos C + ( b + c ) cos A + ( c + a ) cos B is equal to _______. Maharashtra HSC Class 12 · 2023 · Q. 1. ii. Easy Add #22 · Solution of Triangle — Sine, Cosine and Projection Rules
In △ A B C \bigtriangleup ABC △ A B C , if c 2 + a 2 − b 2 = a c c^{2} + a^{2} - b^{2} = ac c 2 + a 2 − b 2 = a c , then ∠ B \angle B ∠ B = ______. Maharashtra HSC Class 12 · 2022 · Q. 1. ii. Moderate Add #23 · Solution of Triangle — Sine, Cosine and Projection Rules
In △ A B C \bigtriangleup ABC △ A B C , if c 2 = a 2 + b 2 = a c c^{2} = a^{2} + b^{2} = ac c 2 = a 2 + b 2 = a c , then ∠ B \angle B ∠ B = ? Maharashtra HSC Class 12 · 2020 · Q. 1. i. Easy Add #24 · Solution of Triangle — Sine, Cosine and Projection Rules
In △ A B C \bigtriangleup ABC △ A B C , if a = 2 , b = 3 a = 2,b = 3 a = 2 , b = 3 and s i n A = 2 3 sinA = \frac{2}{3} s in A = 3 2 , then ∠ B \angle B ∠ B = ? #25 · Applications of Sine, Cosine and Projection Rules
In Δ A B C \Delta ABC Δ A B C , if A = 45 ∘ A = 45^{\circ} A = 4 5 ∘ , B = 60 ∘ B = 60^{\circ} B = 6 0 ∘ then find the ratio of its sides. Maharashtra HSC Class 12 · 2025 · Q. 17 Moderate Add #26 · Applications of Sine, Cosine and Projection Rules
In △ A B C \bigtriangleup ABC △ A B C if a = 13 , b = 14 , c = 15 a = 13,b = 14,c = 15 a = 13 , b = 14 , c = 15 then find the values of (i) s e c A secA sec A (ii) csc A 2 \csc\frac{A}{2} csc 2 A . Maharashtra HSC Class 12 · 2024 · Q. 5 Easy Add #27 · Applications of Sine, Cosine and Projection Rules
In △ A B C \bigtriangleup ABC △ A B C , if a = 18 , b = 24 and c = 30 a = 18,b = 24\text{ and }c = 30 a = 18 , b = 24 and c = 30 then find the value of sin ( A 2 ) \sin\left( \frac{A}{2} \right) sin ( 2 A ) . Maharashtra HSC Class 12 · 2024 · Q. 17 Moderate Add #28 · Applications of Sine, Cosine and Projection Rules
In △ A B C \bigtriangleup ABC △ A B C , prove that: c o s A a + c o s B b + c o s C c = a 2 + b 2 + c 2 2 a b c \frac{cosA}{a} + \frac{cosB}{b} + \frac{cosC}{c} = \frac{a^{2} + b^{2} + c^{2}}{2abc} a cos A + b cos B + c cos C = 2 ab c a 2 + b 2 + c 2 . Maharashtra HSC Class 12 · 2020 · Q. 30 Moderate Add #29 · Applications of Sine, Cosine and Projection Rules
In △ A B C \bigtriangleup ABC △ A B C , if a + b + c = 2 s a + b + c = 2s a + b + c = 2 s , then prove that sin ( A 2 ) = ( s − b ) ( s − c ) b c \sin\left( \frac{A}{2} \right) = \sqrt{\frac{(s - b)(s - c)}{bc}} sin ( 2 A ) = b c ( s − b ) ( s − c ) , with usual notations. Maharashtra HSC Class 12 · 2019 · Q. 15 Moderate Add #30 · Applications of Sine, Cosine and Projection Rules
In △ A B C \bigtriangleup ABC △ A B C , prove that sin ( B − C 2 ) = ( b − c ) a cos ( A 2 ) \sin\left( \frac{B - C}{2} \right) = \frac{(b - c)}{a}\cos\left( \frac{A}{2} \right) sin ( 2 B − C ) = a ( b − c ) cos ( 2 A ) . Maharashtra HSC Class 12 · 2019 · Q. 15 (OR) Moderate Add #31 · Applications of Sine, Cosine and Projection Rules
In △ A B C \bigtriangleup ABC △ A B C , with usual notations prove that ( a − b ) 2 cos 2 ( C 2 ) + ( a + b ) 2 sin 2 ( C 2 ) = c 2 (a - b)^{2}\cos^{2}\left( \frac{C}{2} \right) + (a + b)^{2}\sin^{2}\left( \frac{C}{2} \right) = c^{2} ( a − b ) 2 cos 2 ( 2 C ) + ( a + b ) 2 sin 2 ( 2 C ) = c 2 . Maharashtra HSC Class 12 · 2017 · Q. 1. (A) iii. Moderate Add #32 · Applications of Sine, Cosine and Projection Rules
In △ A B C \bigtriangleup ABC △ A B C , if a = 13 , b = 14 , and c = 15 a = 13,b = 14,\text{ and }c = 15 a = 13 , b = 14 , and c = 15 , then sin ( A 2 ) \sin\left( \frac{A}{2} \right) sin ( 2 A ) = ______. Maharashtra HSC Class 12 · 2016 · Q. 3. (B) i. Moderate Add #33 · Applications of Sine, Cosine and Projection Rules
In △ A B C \bigtriangleup ABC △ A B C with the usual notations, prove that ( a − b ) 2 cos 2 ( C 2 ) + ( a + b ) 2 sin 2 ( C 2 ) = c 2 (a - b)^{2}\cos^{2}\left( \frac{C}{2} \right) + (a + b)^{2}\sin^{2}\left( \frac{C}{2} \right) = c^{2} ( a − b ) 2 cos 2 ( 2 C ) + ( a + b ) 2 sin 2 ( 2 C ) = c 2 . #34 · Inverse Trigonometric Functions and Principal Values
Find the value of sin − 1 ( sin 5 π 3 ) \sin^{-1}\left(\sin\frac{5\pi}{3}\right) sin − 1 ( sin 3 5 π ) #35 · Inverse Trigonometric Functions and Principal Values
Evaluate : cos − 1 ( 1 2 ) + 2 sin − 1 ( 1 2 ) \cos^{-1}\left(\frac{1}{2}\right) + 2\sin^{-1}\left(\frac{1}{2}\right) cos − 1 ( 2 1 ) + 2 sin − 1 ( 2 1 ) Maharashtra HSC Class 12 · 2022 · Q. 2. i. Easy Add #36 · Inverse Trigonometric Functions and Principal Values
Find the principal value of cot − 1 ( − 1 3 ) \cot^{- 1}\left( - \frac{1}{\sqrt{3}} \right) cot − 1 ( − 3 1 ) . Maharashtra HSC Class 12 · 2015 · Q. 1. (A) ii. Easy Add #37 · Inverse Trigonometric Functions and Principal Values
The principal solution of cos − 1 ( − 1 2 ) \cos^{- 1}\left( - \frac{1}{2} \right) cos − 1 ( − 2 1 ) is #38 · Properties of Inverse Trigonometric Functions
The value of sin [ sin − 1 ( − 1 2 ) − 3 sin − 1 ( 3 2 ) ] = \sin\left[\sin^{-1}\left(-\frac{1}{\sqrt{2}}\right) - 3\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\right] = sin [ sin − 1 ( − 2 1 ) − 3 sin − 1 ( 2 3 ) ] = A − 1 2 -\frac{1}{\sqrt{2}} − 2 1 B C D − 3 2 -\frac{\sqrt{3}}{2} − 2 3 #39 · Properties of Inverse Trigonometric Functions
Prove that :
sin − 1 ( 3 5 ) + cos − 1 ( 12 13 ) = sin − 1 ( 56 65 ) \sin^{-1}\left(\frac{3}{5}\right) + \cos^{-1}\left(\frac{12}{13}\right) = \sin^{-1}\left(\frac{56}{65}\right) sin − 1 ( 5 3 ) + cos − 1 ( 13 12 ) = sin − 1 ( 65 56 ) #40 · Properties of Inverse Trigonometric Functions
cos [ tan − 1 ( 1 3 ) + tan − 1 ( 1 2 ) ] = \cos\left[\tan^{-1}\left(\frac{1}{3}\right) + \tan^{-1}\left(\frac{1}{2}\right)\right] = cos [ tan − 1 ( 3 1 ) + tan − 1 ( 2 1 ) ] = _____. #41 · Properties of Inverse Trigonometric Functions
If tan − 1 ( 2 x ) + tan − 1 ( 3 x ) = π 4 \tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4} tan − 1 ( 2 x ) + tan − 1 ( 3 x ) = 4 π then find the value of x x x , where 0 < 3 x < 1 0 < 3x < 1 0 < 3 x < 1 #42 · Properties of Inverse Trigonometric Functions
If tan − 1 ( 2 x ) + tan − 1 ( 3 x ) = π 4 \tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4} tan − 1 ( 2 x ) + tan − 1 ( 3 x ) = 4 π , then x = x = x = _____. Maharashtra HSC Class 12 · 2025 · Q. 16 Moderate Add #43 · Properties of Inverse Trigonometric Functions
Prove that: 2 tan − 1 ( 1 3 ) + cos − 1 ( 3 5 ) = π 2 2\tan^{- 1}\left( \frac{1}{3} \right) + \cos^{- 1}\left( \frac{3}{5} \right) = \frac{\pi}{2} 2 tan − 1 ( 3 1 ) + cos − 1 ( 5 3 ) = 2 π . Maharashtra HSC Class 12 · 2024 · Q. 16 Easy Add #44 · Properties of Inverse Trigonometric Functions
Prove that: tan − 1 ( 1 2 ) + tan − 1 ( 1 3 ) = π 4 \tan^{- 1}\left( \frac{1}{2} \right) + \tan^{- 1}\left( \frac{1}{3} \right) = \frac{\pi}{4} tan − 1 ( 2 1 ) + tan − 1 ( 3 1 ) = 4 π . Maharashtra HSC Class 12 · 2023 · Q. 16 Moderate Add #45 · Properties of Inverse Trigonometric Functions
If − 1 ≤ x ≤ 1 - 1 \leq x \leq 1 − 1 ≤ x ≤ 1 , then prove that sin − 1 x + cos − 1 x = π 2 \sin^{- 1}x + \cos^{- 1}x = \frac{\pi}{2} sin − 1 x + cos − 1 x = 2 π . Maharashtra HSC Class 12 · 2022 · Q. 15 Moderate Add #46 · Properties of Inverse Trigonometric Functions
If 2 tan − 1 ( c o s x ) = tan − 1 ( 2 c o s e c x ) 2\tan^{- 1}(cosx) = \tan^{- 1}(2cosecx) 2 tan − 1 ( cos x ) = tan − 1 ( 2 cosec x ) , then find the value of x x x . Maharashtra HSC Class 12 · 2020 · Q. 16 Moderate Add #47 · Properties of Inverse Trigonometric Functions
Show that: sin − 1 ( 3 5 ) + sin − 1 ( 8 17 ) = sin − 1 ( 77 85 ) \sin^{- 1}\left( \frac{3}{5} \right) + \sin^{- 1}\left( \frac{8}{17} \right) = \sin^{- 1}\left( \frac{77}{85} \right) sin − 1 ( 5 3 ) + sin − 1 ( 17 8 ) = sin − 1 ( 85 77 ) . Maharashtra HSC Class 12 · 2018 · Q. 3. (B) i. Moderate Add #48 · Properties of Inverse Trigonometric Functions
Prove that: sin − 1 ( 3 5 ) + cos − 1 ( 12 13 ) = sin − 1 ( 56 65 ) \sin^{- 1}\left( \frac{3}{5} \right) + \cos^{- 1}\left( \frac{12}{13} \right) = \sin^{- 1}\left( \frac{56}{65} \right) sin − 1 ( 5 3 ) + cos − 1 ( 13 12 ) = sin − 1 ( 65 56 ) . Maharashtra HSC Class 12 · 2017 · Q. 3. (B) ii. Moderate Add #49 · Properties of Inverse Trigonometric Functions
Show that cos − 1 ( 4 5 ) + cos − 1 ( 12 13 ) = cos − 1 ( 33 65 ) \cos^{- 1}\left( \frac{4}{5} \right) + \cos^{- 1}\left( \frac{12}{13} \right) = \cos^{- 1}\left( \frac{33}{65} \right) cos − 1 ( 5 4 ) + cos − 1 ( 13 12 ) = cos − 1 ( 65 33 ) . Maharashtra HSC Class 12 · 2016 · Q. 1. (A) ii. Moderate Add #50 · Properties of Inverse Trigonometric Functions
If sin − 1 ( 1 − x ) − 2 sin − 1 x = π 2 \sin^{- 1}(1 - x) - 2\sin^{- 1}x = \frac{\pi}{2} sin − 1 ( 1 − x ) − 2 sin − 1 x = 2 π then x x x is Studying this with friends? Send them this chapter's questions. It's free for them too.
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